Multivariate T-Distributions and Their Applications

Multivariate t Distributions and Their Applications Almost all of the results available in the literature on multivaria...

Author: Samuel Kotz | Saralees Nadarajah

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Multivariate t Distributions and Their Applications Almost all of the results available in the literature on multivariate t distributions published in the last 50 years are now collected together in this comprehensive volume. Because these distributions are becoming more prominent in many applications, this book is a must for any serious researcher or consultant working in multivariate analysis and statistical distributions. Much of this material has never appeared in book form. The first part of the book emphasizes theoretical results of a probabilistic nature. In the second part of the book, these are supplemented by a variety of statistical aspects. Various generalizations and applications

are dealt with in the final chapters. The material on estimation and regression models is of special value for practitioners in statistics and economics. A comprehensive bibliography of more than 350 references is included.

Samuel Kotz is Professor and Senior Research Scholar in the Department of Engineering Management and Systems Engineering at George Washington University. In addition to holding many distinguished vis-

iting positions at prominent universities, he has authored a number of books and more than 100 articles, is Editor-in-Chief and founder of the Encyclopedia of Statistical Sciences, and holds three honorary doctorates.

Saralees Nadarajah is Professor of Mathematics at the University of South Florida. He has made important contributions to distribution theory, extreme value theory and its applications in environmental modeling, branching processes, and sampling theory. He has authored more than 80 papers and two books.

Multivariate t Distributions and Their Applications Samuel Kotz George Washington University

Saralees Nadarajah University of South Florida

CAMBRIDGE UNIVERSITY PRESS

PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE

The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS

The Edinburgh Building, Cambridge C132 2RU, UK 40 West 20th Street, New York, NY 10011-4211, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia Ruiz de Alarcon 13, 28014 Madrid, Spain Dock House, The Waterfront, Cape Town 8001, South Africa

http://www.cambridge.org

O Samuel Kotz and Saralees Nadarajah 2004

This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2004

Printed in the United States of America Typeface Computer Modern 10/13 pt.

System LAT C 26 [AU]

A catalog record for this book is available from the British Library.

Library of Congress Cataloging in Publication Data Kotz, Samuel.

Multivariate t distributions and their applications / Samuel Kotz, Saralees Nadarajah. p. cm.

Includes bibliographical references and index. ISBN 0-521-82654-3

1. Multivariate analysis. 2. Distribution (Probability theory) I. Nadarajah, Saralees. 11. Title.

QA278.K635 2004 2003055353 519.5'35-dc2l ISBN 0 521 82654 3 hardback

Contents

List of Illustrations Preface

page ix xi

1 Introduction 1.1 1.2 1.3

1

Definition Representations Characterizations

1

2

1.4 A Closure Property 1.5 A Consistency Property 1.6 Density Expansions 1.7 Moments 1.8 Maximums 1.9 Distribution of a Linear Function

1.10 Marginal Distributions 1.11 Conditional Distributions 1.12 Quadratic Forms 1.13 F Matrix 1.14 Association 1.15 Entropy 1.16 Kullback-Leibler Number 1.17 Renyi Information 1.18 Identities 1.19 Some Special Cases

2 The Characteristic Function

7 8 9 9 10 12 15 15 16 19 20 20 21 23 26 30 33

2.1 2.2

Sutradhar's Approach Joarder and Ali's Approach

36 36 38

2.3

Levy Representation

41

3 Linear Combinations, Products, and Ratios 3.1 3.2 3.3

Linear Combinations Products Ratios

44

44 52 56

v

vi

Contents

4 Bivariate Generalizations and Related Distributions Owen's Noncentral Bivariate t Distribution Siddiqui's Noncentral Bivariate t Distribution Patil and Liao's Noncentral Bivariate t Distribution Krishnan's Noncentral Bivariate t Distribution Krishnan's Doubly Noncentral Bivariate t Distribution Bulgren et al.'s Bivariate t Distribution Siotani's Noncentral Bivariate t Distribution Tiku and Kambo's Bivariate t Distribution Conditionally Specified Bivariate t Distribution 4.10 Jones' Bivariate t Distribution 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9

5 Multivariate Generalizations and Related Distributions Kshirsagar's Noncentral Multivariate t Distribution Miller's Noncentral Multivariate t Distribution Stepwise Multivariate t Distribution Siotani's Noncentral Multivariate t Distribution Arellano-Valle and Bolfarine's Generalized t Distribution Fang et al.'s Asymmetric Multivariate t Distribution Gupta's Skewed Multivariate t Distribution Sahu et al.'s Skewed Multivariate t Distribution Azzalini and Capitanio's Skewed Multivariate t Distribution 5.10 Jones' Skewed Multivariate t Distribution 5.11 Matrix-Variate t Distribution 5.12 Complex Multivariate t Distribution 5.13 Steyn's Nonnormal Distributions 5.14 Inverted Dirichlet Distribution 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9

6 Probability Integrals Dunnett and Sobel's Probability Integrals Gupta and Sobel's Probability Integrals John's Probability Integrals 6.4 Amos and Bulgren's Probability Integrals 6.5 Steffens' Noncentral Probabilities 6.6 Dutt's Probability Integrals 6.7 Amos' Probability Integral 6.8 Fujikoshi's Probability Integrals 6.9 Probabilities of Cone 6.10 Probabilities of Convex Polyhedra 6.11 Probabilities of Linear Inequalities 6.12 Maximum Probability Content 6.13 Monte Carlo Evaluation 6.1 6.2 6.3

7 Probability Inequalities 7.1 7.2

Dunnett and Sobel's Probability Inequalities Dunn's Probability Inequalities

63 63 66 68 69 71 72 73 74 76 80

87 87 90 90 93 94 97 98 102 103 105 112 119 120 126 127 127 131 135 137 139 140 143 144 145 148 158 160 161 165 165 169

Contents

vii

7.3 Halperin's Probability Inequalities 7.4 Sidak's Probability Inequalities 7.5 Tong's Probability Inequalities

170

8 Percentage Points

Dunnett and Sobel's Percentage Points Krishnaiah and Armitage's Percentage Points Gupta et al.'s Percentage Points Rausch and Horn's Percentage Points Hahn and Hendrickson's Percentage Points Siotani's Percentage Points 8.7 Graybill and Bowden's Percentage Points 8.8 Pillai and Ramachandran's Percentage Points 8.9 Dunnett's Percentage Points 8.10 Gupta and Sobel's Percentage Points 8.11 Chen's Percentage Points 8.12 Bowden and Graybill's Percentage Points 8.13 Dunnett and Tamhane's Percentage Points 8.14 Kwong and Liu's Percentage Points 8.15 Other Results 8.1 8.2 8.3 8.4 8.5 8.6

9 Sampling Distributions

171

172

174 174 175 176

176 177 177 178

180 180 181

182

183 183

187 187 191

9.1

Wishart Matrix

191

9.2

Multivariate t Statistic

198

9.3 Hotelling's T2 Statistic 9.4 Entropy and Kullback-Leibler Number

10 Estimation 10.1 Tiku and Kambo's Estimation Procedure 10.2 ML Estimation via EM Algorithm 10.3 Missing Data Imputation 10.4 Laplacian T-Approximation 10.5 Sutradhar's Score Test 10.6 Multivariate t Model 10.7 Generalized Multivariate t Model 10.8 Simulation

11 Regression Models 11.1 Classical Linear Model 11.2 Bayesian Linear Models 11.3 Indexed Linear Models 11.4 General Linear Model 11.5 Nonlinear Models

12 Applications 12.1 Projection Pursuit 12.2 Portfolio Optimization

199

204 207

207 210 212

214 215

219 222 223 228 228

233 235 237 239 241

241

243

Contents

viii

12.3 Discriminant and Cluster Analysis 12.4 Multiple Decision Problems 12.5 Other Applications

244 245 246

References Index

247 269

List of Illustrations

1.1

1.2

1.3 1.4 1.5 1.6 3.1

Joint contours of (1.1) with degrees of freedom v = 1, zero means, and correlation coefficient p 0.8,0.6,.. . , -0.6, -0.8 page 3 Joint contours of (1.1) with degrees of freedom v = 2, zero means, and correlation coefficient p = 0.8,0.6,.. . , -0.6, -0.8 4 Joint contours of (1.1) with degrees of freedom v = 10, zero means, and correlation coefficient p = 0.8,0.6,.. . , -0.6, -0.8 5 Joint contours of (1.1) with degrees of freedom v = 30, zero means, and correlation coefficient p = 0.8,0.6,.. . , -0.6, -0.8 6 Mutual information, (1.31), for p = 2 26 Mutual information, (1.31), for p = 4 27 Densities of the t-ratio distribution (3.11) for (mm, my, v) _ (0, 0, 1), (0, 0, 30), (1, 3, 1), (1, 3, 30), (3,1,1), (3, 1, 30), (3, 3, 1), and (3, 3, 30)

4.1

4.2

Jones' bivariate skew t pdf (4.33) for (a) vi = 2 and v2 = 3; and (b) vi = 2 and v2 = 20 Jones' bivariate skew t pdf (4.40) for (a) a = 1, b = 1, and

58

83

c = 1; (b) a = 3, b = 4, and c = 5; (c) a = 5, b = 1, and

c=1;and(d)a=1,b=5,andc=1 5.1

85

Fang et al.'s asymmetric t pdf (5.16) in the bivariate case

(a)m=2,mi =10,m2=10,andr12=0; (b)m=2,

m1 = 10, m2 = 2, and r12 = 0; (c) rn = 2, m1 = 10,

m2=10,and r12=0.5;and (d)m=2,m1=10,m2=10, 5.2

and r12 = 0.9 Jones' skewed multivariate t pelf (5.28) for p = 2 and (a)

a=6,v=3,andc=2; and (b) a = 2, v = 3, and c = 6 5.3 5.4

99 107

Jones' skewed multivariate t pdf (5.29) for p = 2 and (a) v = 1; and (b) v = 20 108 Jones' skewed multivariate t pdf (5.32) for p = 2 and (a) vo = 2, v1 = 4, and v2 = 4; (b) vo = 2, vi = 20, and v2 = 1; (c) vo = 2, vi = 1, and v2 = 20; and (d) vo = vi = v2 = 2 112

List of Illustrations 5.5

Steyn's bivariate pdf corresponding to (5.46) for t3 = 0 and (a) #1 = 0.8, n2 = -0.4, and r12 = 0.2; (b) i1 = 0.8, K2 = -0.4, and r12 = 0.8; (c) ic1 = -0.4, r-2 = 0.8, and r12 = 0.2; and (d) n1 = -0.4, K2 = 0.8, and r12 = 0.8

6.1

The sets Ar(c) and E(c) f1 {II z II= r} in two dimensions

124 146

Preface

Multivariate t distributions have attracted somewhat limited attention of researchers for the last 70 years in spite of their increasing importance in classical as well as in Bayesian statistical modeling. These distributions have been perhaps unjustly overshadowed - during all these years - by

the multivariate normal distribution. Both the multivariate t and the multivariate normal are members of the general family of elliptically symmetric distributions. However, we feel that it is desirable to focus on these distributions separately for several reasons: Multivariate t distributions are generalizations of the classical univariate Student t distribution, which is of central importance in statistical inference. The possible structures are numerous, and each one possesses special characteristics as far as potential and current applications are concerned.

Application of multivariate t distributions is a very promising approach in multivariate analysis. Classical multivariate analysis is soundly and rigidly tilted toward the multivariate normal distribution while multivariate t distributions offer a more viable alternative with respect to real-world data, particularly because its tails are more realistic. We have seen recently some unexpected applications in novel areas such as cluster analysis, discriminant analysis, multiple regression, robust projection indices, and missing data imputation. Multivariate t distributions for the past 20 to 30 years have played a crucial role in Bayesian analysis of multivariate data. They serve by now as the most popular prior distribution (because elicitation of prior information in various physical, engineering, and financial phenomena is closely associated with multivariate t distributions) and generate meaningful posterior distributions. This diversity and the apparent xi

xii

Preface

ease of applications require careful analysis of the properties of the distribution in order to avoid pitfalls and misrepresentation.

The compilation of this book was a somewhat daunting task (as our Contents indicates). Indeed, the scope of the multivariate t distributions is unsurpassed, and, although there are books dealing with multivariate continuous distributions and review articles in the Encyclopedia of Statistical Sciences and Biostatistics, the material presented in these sources is quite limited. Our goal was to collect and present in an organized and user-friendly manner all of the relevant information available in the literature worthy of publication. It is our hope that the readers - both novices and experts - will find the book useful. Our thanks are due to numerous authors who generously supplied us with their contributions and to Lauren Cowles, Elise Oranges and Lara Zoble at Cambridge University Press for their guidance. We also wish to thank Anusha Thiyagarajah for help with editing. Samuel Kotz Saralees Nadarajah

1

Introduction

1.1 Definition There exist quite a few forms of multivariate t distributions, which will be discussed in subsequent chapters. In this chapter, however, we shall describe the most common and natural form. It directly generalizes the univariate Student's t distribution in the same manner that the multivariate normal distribution generalizes the univariate normal distribution. A p-dimensional random vector X = (X1, ... , XP)T is said to have the p-variate t distribution with degrees of freedom v, mean vector µ, and correlation matrix R (and with E denoting the corresponding covariance matrix) if its joint probability density function (pdf) is given by

f(x) =

r ((v + p)/2) (xv)p/2r (v/2) IR,I1/2

1

T

1

Ll + v (x - µ) R- (x - µ)J

(v+P)/2

(1.1)

The degrees of freedom parameter v is also referred to as the shape parameter, because the peakedness of (1.1) may be diminished, preserved, or increased by varying v (see Section 1.4). The distribution is said to be central if µ = 0; otherwise, it is said to be noncentral.

Note that if p = 1, µ = 0, and R = 1, then (1.1) is the pdf of the univariate Student's t distribution with degrees of freedom v. These univariate marginals have increasingly heavy tails as v decreases toward unity. With or without moments, the marginals become successively less peaked about 0 e as v 11. If p = 2, then (1.1) is a slight modification of the bivariate surface of Pearson (1923). If v = 1, then (1.1) is the p-variate Cauchy distribution. If (v+p)/2 = m, an integer, then (1.1) is the p-variate Pearson type VII 1

2

Introduction

distribution. The limiting form of (1.1) as v -> oo is the joint pdf of the p-variate normal distribution with mean vector p and covariance matrix E. Hence, (1.1) can be viewed as an approximation of the multivariate

normal distribution. The particular case of (1.1) for µ = 0 and R = Ir is a mixture of the normal density with zero means and covariance matrix vIr - in the scale parameter v. The class of elliptically contoured distributions (see, for example, Fang et al., 1990) contain (1.1) as a particular case. Also (1.1) has the attractive property of being Schurconcave when elements of R satisfy ri.7 = p, i $ j (see Marshall and Olkin, 1974). Namely, if a and b are two p-variate vectors with compo> bp, and nents ordered to achieve al > a2 > > ap and b1 > b2 > , p - 1 and ai < E if this ordering implies bi for k = 1,2,.. . 1 1 EP 1 ai < EP bi, then (1.1) satisfies f (a) > f (b). In Bayesian analyses, (1.1) arises as: (1) the posterior distribution of the mean of a multivariate normal distribution (Geisser and Cornfield, 1963; see also Stone, 1964); (2) the marginal posterior distribution of the regression coefficient vector of the traditional multivariate regression model (Tiao and Zellner, 1964); (3) the marginal prior distribution of the mean of a multinormal process (Ando and Kaufman, 1965); (4) the marginal posterior distribution of the mean and the predictive distribution of a future observation of the multivariate normal structural model (Fraser and Haq, 1969); (5) an approximation to posterior distributions arising in location-scale regression models (Sweeting, 1984, 1987); and (6) the prior distribution for set estimation of a multivariate normal mean (DasGupta et al., 1995). Additional applications of (1.1) can be seen in the numerous books dealing with the Bayesian aspects of multivariate analysis.

1.2 Representations If X has the p-variate t distribution with degrees of freedom v, mean vector µ, and correlation matrix R, then it can be represented as If Y is a p-variate normal random vector with mean 0 and covariance

matrix E, and if vS2/Q2 is the chi-squared random variable with degrees of freedom v, independent of Y, then

X = S-1y + µ.

(1.2)

This implies that X I S = s has the p-variate normal distribution with mean vector µ and covariance matrix (1/82)E.

1.2 Representations

3

WAM

rha:OAM:1

M

-3

4

4

4

-1

-1

0

1

0

1

2

2

3

3

4

4

4

4

-1

0

0

2

1

1

2

3

3

4

4

4

4

-1

-1

0

0

1

1

2

3

2

3

2

3

t*-04L1 M

N

-3

-2

-1

0

1

2

3

-3

-2

-1

0

1

2

3

4

4

-1

0

1

Fig. 1.1. Joint contours of (1.1) with degrees of freedom v = 1, zero means, and correlation coefficient p = 0.8,0.6..... -0.6, -0.8

Introduction

4

M

N

0

-3

-2

-1

0

1

2

3

-3

4

4

-1

0

1

0

I

-3

-2

W.W

dMW

4

-1

2

3

4

4

-1

0

1

-1

0

1

2

3

2

3

2

3

rt*-01n 2

2

3

4

4

-1

0

1

b.4u2

ft:-0.4p4

n

n

-3

3

-1

0

1

2

3

4

4

-1

0

1

2

3

4

-2

-1

0

1

Fig. 1.2. Joint contours of (1.1) with degrees of freedom v = 2, zero means and correlation coefficient p = 0.8, 0.6, ... , -0.6, -0.8

5

1.2 Representations

rha'6 Vlo

3

rla

-1

ri

nth10

4

rl

3

-2

-1

rl

-0.4,rtw10

3

3

-1

0

1

W,4,rIU.-1O

2

3

3

3

0,0, eo

0

1

rl

2

3

3

2

1

-1

0

2

3

3

-2

-f

1

2

3

-0,41t10

0

rtftA

-0,6pit10

0

-1

0

1

2

3

to

1

2

3

Fig. 1.3. Joint contours of (1.1) with degrees of freedom v = 10, zero means, and correlation coefficient p = 0.8,0.6,.. . , -0.6, -0.8

Introduction

6

w4a

4

4

-1

0

1

W.W

2

3

-3

4

-1

0

1

W.W

2

3

4

-2

-1

0

1

2

3

2

3

2

3

r io:41A 30

4

4

-2

4

-1

-1

0

0

1

1

2

2

3

3

4

4

4

4

-1

-1

0

0

1

1

2

2

3

3

4

4

4

4

-1

0

rh

-l2 w

-1

0

1

1

Fig. 1.4. Joint contours of (1.1) with degrees of freedom v = 30, zero means, and correlation coefficient p = 0.8,0.6,..., -0.6, -0.8

1.3 Characterizations

7

If V'/2 is the symmetric square root of V, that is,

V1/2V1/2 = V ,,, Wp (R-1, v + p - 1) ,

(1.3)

where Wp(E,n) denotes the p-variate Wishart distribution with degrees of freedom n and covariance matrix E, and if Y has the p-variate normal distribution with zero means and covariance matrix vIp (Ip is the p-dimensional identity matrix), independent of V, then

X=

1Y

(V1/2)

+µ

(1.4)

(Ando and Kaufman, 1965). This implies that X I V has the p-variate normal distribution with mean vector µ and covariance matrix vV-1.

1.3 Characterizations From representation (1.2) it easily follows for any a ; 0 that X has the joint pdf (1.1) if and only if X I S2 = s2 ' N (µ, s-2 E) .

q

(aTEa)

-112

aT (X - µ) I S2 = s2 - 1 / 2 aT

N (0,8-2)

(X - µ) ,., t,,,

and this is one of the earliest characterization results given in Cornish (1962). This result can also be obtained by using the representation (1.4): X has the joint pdf (1.1) if and only if

XIV - N(p,vV-1)
(aTEa)-1/2 aT (X - µ) I V - N (0, v

C*

(aTEa) -1/2 aT (X - µ) ,., tv,

(aTV- 'a) / (aTEa))

as noted by Lin (1972). Lin (1972) obtained two further characterizations using the representation (1.2). Let vS2 _ x2 and let X1, X2, ... , Xp be conditionally independent continuous random variables symmetrically distributed with E(Xk S2 = s2) = µk and Var(Xk S2 = s2) _ 0.2./s2 < oo for k = 1,. .. , p. Then the following characterizations are valid I

I

(X1, X2, ... , Xp)T has the joint pdf (1.1) with mean vector µ, covari-

ance matrix D, and degrees of freedom v if and only if

(Xk - µk)2 k=1

PL72

Fp,v

Introduction

8

where D is a p x p diagonal matrix with its kth diagonal element equal to 01'

In the special case Q2 = 01 2 for all k and the conditional pdf of Xk S2 = s2 is positive and differentiable for all x E R, (X1, X2i ... , X ,)T has the joint pdf (1.1) with zero means, covariance matrix a2IP, and degrees of freedom v if and only if the joint pdf of X1, X2,. .. , Xr is a function of x1 + x2 + + xP only.

1.4 A Closure Property Consider Studentizing transformations T : In -4 Ik, depending on matrices A(n x k), B(n x v) and 1Z(n x n), given by T (X)

_ fATX

1IB XII

such that ATfIB = 0. Jensen (1994) established that the class of multivariate t distributions is closed under the transform T(.). Specifically, assume ATA = Ik, BTB = I, and X is distributed according to (1.1) with zero means, correlation matrix In, and degrees of freedom m. Under these assumptions, Jensen showed that T(X) is also distributed according to (1.1) with zero means, correlation matrix Ik, and degrees of freedom v.

Jensen (1994) also studied the concentration properties of (1.1) via peakedness by varying its parameters. If X is multivariate normal, then the transformation X -4 T(X) diminishes the peakedness. If, on the other hand, X is distributed according to (1.1) with mean vector µ1n, covariance matrix o 2In, and degrees of freedom m, then the transformation is peakedness-enhancing for all m < v. If m > v > 2, then

the transformation serves to increase variances. For any m > v > 0 the marginal distributions are less peaked after T(X) than before in the sense of Birnbaum (1948). If m = v, then the marginals are identical before and after T(X), thus exhibiting identical tail behavior. If v > m then marginals are more peaked (in the sense of Birnbaum, 1948) after applying T (X) than before; and if v > m > 2, then T (X) serves as a variance-diminishing transformation.

1.5 A Consistency Property

9

1.5 A Consistency Property A random vector X = (X1, ... , XP) T is said to have the spherical distribution if its joint pdf can be written in the form

where g(.) is referred to as the density generator. The p-variate t pdf (1.1) with it = 0 and E = Ip is spherical because in this case, 1 + u -(p+v)/2

F ((v +p)/2)

=

9(u)

(V)

(xv)P/2F (v/2)

Other examples of spherical distributions include the multivariate normal and the multivariate power exponential. A spherical distribution is said to possess the consistency property if

r

P+1

O

P

xi p dxp+1

g

xz P

g

(1.6)

i=1

(i=1

for any integer p and almost all x E RP. This consistency property ensures that any marginal distribution of X also belongs to the same spherical family. Kano (1994) provided several necessary and sufficient conditions for a spherical distribution to satisfy (1.6). One of the them is that g must be a mixture of normal distributions; specifically, there exists a random variable Z > 0, unrelated to p, such that, for any p,

= f(-_)

.f (u I P)

z

p

l

exp (__2 ) F(dz),

where F(.) denotes the cumulative distribution function (cdf) of Z. Since the multivariate t is a mixture of normal distributions (see (1.2)), it follows that it must have the consistency property. Other distributions that have the consistency property include the multivariate normal and the multivariate Cauchy. Distributions that do not share this property include the multivariate logistic, multivariate Pearson type II, multivariate Pearson type VII , and the multivariate Bessel.

1.6 Density Expansions Fisher (1925) and later Dickey (1967a) provided expansions of the pdf f (x)

-

1 + x2

F((v + 1)/2) 7rvF (v/2)

1.

V

(v+1)/2

}

Introduction

10

of the univariate Student's t distribution. The expansion in the latter paper involves Appell's polynomials, and hence recurrence schemes are available for its coefficients. Specifically, 00

f(x) =

exp ( 12

Qk )k=0 >

(_

vxz)

12

(1 + v),

/f

where k-1

Qk(t) = Pk(t) -

EQ,(t)Pk-l(r) 1=0

Here, Pk(t) are polynomials (in powers of t) satisfying

Pk(t)(1 +

v)-k

=

11- 1+

v)

exp(-t)

and Pk(F) denotes the polynomial Pk(t) with the powers t' replaced by F(r + 1/2). Dickey (1967a) also provided an analog of (1.7) for the multivariate t pdf (1.1). It takes the same form as (1.7) with x 2 replaced by (x- µ)T R-1(x - µ), v+ 1 replaced by v + p, and with (1.8) replaced by 1

(t)

=

Pk (t)

kQk -1

- F(P/2) EQc (t)Pk-c (rr),

where FP indicates the substitution of F(r + p/2) for t'.

1.7 Moments Since Y and S in (1.2) are independent, the conditional distribution of (Xi, Xj), given S = s, is bivariate normal with means (µi, µ0, common variance a2/s2, and correlation coefficient rid. Thus, E (Xi)

= E [E (Xij S = s)] E (pi) µi

To find the second moments, consider the classical identity Coo (Xi, Xj)

= E [Cov (Xi, Xj) I S = s] +Cov [E (Xil S = s) E (Xjj S = s)]

1.7 Moments

11

for all i, j = 1, ... , p. Clearly, one has E [Cov (Xi, Xj) I S = s]

= o2rzjE

SZ 1)

and

Cov [E (XZIS = s) E (Xj S = s)] =

0.

If v > 2, then E(1/S2) exists and is equal to v/{o2(v - 2)}. Thus, by choosing i = j and i < j, respectively, one obtains

Var(X )

=

V

v - 2

and

Cov (Xi, Xj) =

v

v

2rZj.

Hence the matrix R is indeed the correlation matrix as stated in definition (1.1). In the case where µ = 0, the product moments of X are easily found by exploiting the independence of Y and S in (1.2). One obtains J"'rl,rz,..., rp j=1

E S-r (llYr3)] j-1 1

Q-rvrl2E II Yr'JI E j-1 provided that r

[xV r]

+ rp < v12. In the special case where

r1 + r2 +

Y1,. .. , Yp are mutually independent, one obtains p

,Url,rz,..., rp

Q-rvr/2E [XV r] II E [Yjr'] j=1

If anyone of the rj's is odd, then the moment is zero. If all of them are even, then r12fjp=1 llri,rz,...,rp

=

(v - 2)(v - 4) ... (v - r)

1)}

Introduction

12

In particular, v

=

µ2,0,...,o

v > 2,

v - 2' 3v2

4,

(v - 2)(v - 4) µ2,2,0,...,o

v2

-

(v

v > 4,

- 2)(v - 4)'

and

=

/12,2,2,0,...,0

v3

v>6.

(v - 2)(v - 4)(v - 6)'

1.8 Maximums Of special interest are the moments of Z = max(Xi,... Xp) when XT = (X1,. .. , Xp) has the t pdf (1.1) with the mean vector µ and ,

covariance matrix E. These moments have applications in decision theory, particularly in the selection and estimation of the maximum of a set of parameters. It also has applications in forecasting. The problem of finding the moments of Z has been considered by Raiffa and Schlaifer (1961), Afonja (1972), and Cain (1996). Raiffa and Schlaifer (1961) provided an expression for E(Z - 0) for

the case where p = 3 and µ = 01p (where 1p denotes a vector of l's). Afonja (1972) generalized this for the general case of unequal means, variances, and correlations. We mention later a particular case of this result for µ = 01p. Let Op(y; R) denote a p-dimensional normal pdf with zero means, unit variances, and correlation matrix R. Also let Ri denote a p x p matrix with its (j, j')th element equal to ri,jj,, where ri,jj, (j, j' 54 i) is the correlation between (Xi, -Xj) and (Xi, -Xj,) and ri,ij = corr(Xi, Xi - Xj). Then the kth moment of Z is given by E (Zk)

_

1 p k kl Bk-7 ( r (v12)( \2(v - 2) (j/ j=0 2_1

where

µj (yi)

=J 0

foo"

r

... J0

I

))'/2

V2

) µ (yi)

yi 0p (y; Ri)

0 dypdyp-1...dyi...dy2dy1

(1.10)

1.8 Maximums

13

is the marginal moment (up to a constant) of truncated normal variates.

The mean and variance can be derived easily from this formula. For example,

E(Z) = 9+{E(W) -9}F (-±)

/r (2)

,

where W = max(Yi, ... , Y,) for a p-variatenormalll random vector yT = (Y1, ... , YY) with means equal to 9 and covariance matrix (vl (v - 2)) E. Afonja (1972) showed further that n

E(W)

+

Vvv

2

iipi (yi) i1 -

where pi (yi) is given by (1.10) for j = 1. More recently, Cain (1996) considered two forecasts Fl and F2 of a future variable Y where the forecast errors Xl = Fi -Y and X2 = F2 - Y are assumed to have the bivariate t distribution with means (A1, A2), variances (o , o ), correlation coefficient p, and degrees of freedom v >

2. Cain was interested in the maximum Z = max(Xi, X2) of the two forecast errors and whether this nonlinear function could be useful as a component of a linear combination forecast. It was shown that the pdf of Z can be written as the sum =

f(z)

fi(z) + f2 (Z),

where

f, (z)

=

r

vv2tu \ vv 2aipZ/ 1+vlz-`k

l+

L

xT

ok

z=& l

- p a;

J

1-p2 v-2+ (z° )

2

for k = 3 - j, j = 1, 2. Here, t and T are, respectively, the pdf and the cdf of the Student's t distribution with degrees of freedom v. Integration by parts yields that

E(Z) = pi

J

fl(z)dz + p2

f2(z)dz +

FOO 00

Var(Z)

= Ql F fl(z)dz 00

f2(z)dz

a2 J 00

T

,

Introduction

14

ao

+ (µl - µ2 )2

fl(z)dzj

J

f2(z)dz

ao

µl

(µl - p2) tv_2

T

00

µ2

J

/

-T (µl - µ2)

C µ1

T

µ'2

f2(z)dz 00

fl z dz

J

(µl-µ2)(Q2-Q1µl-µ2+

T(v - 2) t

T

and 00

Cov (Z, XI) = Q1 f7fl(z)dz+P0'0`2f f2(z)dz 00

0

+ (µl

- µ2)(o -

µ1 - µ2

(

T(v - 2)

T

)

where T = o uu2 - 2pala2. The two integrals in the above expressions can be evaluated as 00

Al

7'v

fl(z)dz

A2

v-2

C

0J0

and

f2(z)dz

=

1 -T (µ1 - /t2

v

T

0o

2

The expression for Cov(Z, X2) can be obtained by switching the subscripts 1 and 2. As v -> oo, the above expressions can be reduced by replacing by and respectively. On the other and extreme, as v -+ 2+, the expressions could be reduced by using the fact

that

IxI

ifx=0,

0,

lim ti_2(x) v*2+v-2

1/2, ifx54 0,

and

1/2,

ifx > 0, ifx = 0,

0,

ifx<0.

1,

lim

v U 2x)

=

This suggests that the results for the maximum of bivariate t distributed errors may be materially different from those for bivariate normal errors.

1.9 Distribution of a Linear Function

15

Cain (1996) also investigated to see whether the maximum Z can provide information additional to that of F1 and F2 in forecasting Y via a linear combination of the form F = a + /31F1 + /32 F2 + ryM with

/31 + /2 + y = 1. Cain showed that the mean squared error of F is minimized when ^y = 0 and hence that M is linearly dominated by F1 and F2. Similar calculations reveal that the mean forecast (Fl + F2)/2 dominates M if and only if either pi = µ2 or al = a2. Evidently further investigations are in order (to consider, for example, the case of more than two forecasts).

1.9 Distribution of a Linear Function If X has the p-variate t distribution with degrees of freedom v, mean vector p, and correlation matrix R, then, for any nonsingular scalar matrix C and for any a, CX + a has the p-variate t distribution with degrees of

freedom v, mean vector Cp+a, and correlation matrix CRCT. This result is of importance in applications and is similar to the corresponding result for the multivariate normal distribution.

1.10 Marginal Distributions Let X possess the p-variate t distribution with degrees of freedom v, mean vector µ, and correlation matrix R. Consider the partitions

X=

(1.11)

(),

(1.12) and

R =

R12 R22

RR211 1

(1.13)

where X1 is pl x 1 and R11 is pi x pl. Then X1 has the p1-variate t distribution with degrees of freedom v, mean vector µl, correlation matrix R11, and with the joint pdf given by

r((v+pl)/2) f (xl) = (irv)P'/2r (v/2) JR11I1/2 x 1+

I -(xl V

(v+pi)/2 -µl)T

R111

(xl -IL1 )

16

Introduction

Moreover, X2 also has the (p - pi )-variate t distribution with degrees of freedom v, mean vector u2, correlation matrix R22, and with the joint pdf given by F ((v + P - Pi)/2)

f (X2)

(irv)pl/2r (v/2) JR2211/2 rr

+

x [1

(x2 -112 )T R_ 22 (x2 -

ll

(v+p-pl)/2

1.11 Conditional Distributions Several interesting properties have been obtained for conditional pdfs of the multivariate t distribution. If X has the central p-variate t distribution with degrees of freedom v and correlation matrix R, it then follows from Section 1.10 that the conditional pdf of X2 given Xi is given by IR11I1/2

I'((v +p)/2)

f (x2 I xi=

(y7r)pl/2r((v+p1)/2)

JRI1/2

[1 + (1/v)xT Ril xi] (v+p')/2 [1 + (1/v)xTR-1x](v+p)/2

X

(1.14)

Since

=

IRI

JR11I IR22 - R21R111R121

and

= x1 1ixi +x2-1 R22 1x2.1,

xTR-x

where x2.1

= x2 - R21R1i xi

and R22.1

= R22 - R21Rj11R12,

one can rewrite (1.14) as .f (x2 I xi)

_

r ((v + p) /2)

{(v +

pi)7r}(p-p1)/2

X

X

1

1+ v +p1 r L

r ((v +pi)/2)

((v+pi)/v)xaiR221x2.1l -v+p2 1+ (1/v)xTRjiixi i

(v + pi) /y 111xi 1 + (1/v)x1 R1-11x11

I

1(p-pl )/2

(1.15)

1.11 Conditional Distributions

17

Landenna and Ferrari (1988) noted that this conditional pdf is not a (p - p1)-variate t unless the values of x1 are ±1. For example, consider the special case of (1.15) for R = I. In this case, (1.15) becomes f (x2

I' ((v +p)/2)

xl) = 7f(P-Pi)/2F ((v

+pl)/2)

/I V + EPl 1

E

EPi j=1

/2

P

1

V +

(P-Pi

j=Pi +1

(1.16)

When xj = ±1, j = 1, 2, :.. pl, (1.16) reduces to .f (x2 I xl)

F ((v +p)/2)

=

7r(P-Pi)/2F ((v +pl)/2) (v + pl)(P-Pi)/2 P

X

1+

Ex V + pl j=Pi+1

which is the joint pdf of a central (p - pi)-variate t distribution with degrees of freedom (v+pl) and correlation matrix IP-P, . Landenna and Ferrari (1988) also described the manner in which the probabilities of the conditional pdf (1.15) can be expressed in terms of the probabilities of x2 conditioned on x1 taking the values ±1. The form of the conditional pdf (1.15) also suggests that

Yl = X1

(1.17)

and

Y2

1 + 1 XTRi i1Xl

(X2 - R2lR1 ilXl) (1.18)

are independent, that Yl has the central pl-variate t distribution with degrees of freedom v and correlation matrix R11, and that Y2 has the central (p - pl )-variate t distribution with degrees of freedom v +p1 and correlation matrix R22.1. From this observation, it follows easily that

the conditional expectation of X2 given X1 is linear and that E(X2 Xl = x1) = R21Rjllxl. In particular, 1 P-1

E(Xp1X1 =x1,...,XP-1 =xP-1)

r*PP j=0

rjpxj

Introduction

18

and

Var (Xp J X, = xl,... , Xp-1 = xp-1)

_

1

v

r* v +P PP

f

1

-

1 +

3

*

rjk

,k_0

r*

xjxk

(1.19)

pp

,

where r*k is the (j, k)th element of R-1 (Bennett, 1961). It is illuminating to compare the conditional variance (1.19) with the value 1/rPp corresponding to the conditional variance of the multivariate normal distribution. Siotani (1976) generalized the result of (1.17)-(1.18) by splitting X

into more than two sets of variates. Let

X =

(1.20)

and

R =

/ R11

R12

... Rlk

R21

R22

R2k

Rkl

Rk2

...

Rkk

where X, is p1 x 1 for l = 1 , 2, ... , k and RI,, is pi x p,,,, for l = 1, 2, ... , k, m = 1, 2, ... , k. Clearly P1 +P2 + +Pk = p. Introducing the notations (1.21)

X(c)

(1.22)

=

..

R11

R12

R21

R22

...

R11

R12

...

R1, R21

R11

(1.23)

1.12 Quadratic Forms

19

R1,1+1

R2

_

R(1+1)

,

1+1

(1.24)

(1)

R1,1+1

and

Rl+l,l+l.(1)

=

- R(l)

R1+1,1+1

1)TR(l1

R(l) 1)

(1.25)

Siotani showed that

Yl = X1 and 1X(1)1-

Y1+1

=

v

ql

v X

1/2

(1 + vX (l) R,

(x(l+l)

- R(1+ 1)T R_1X(l) I

for 1 = 1, ... , k - 1 are independent, that Yl has the central pl-variate t distribution with degrees of freedom v and correlation matrix R11, and that Y1+1 has the central p1+1-variate t distribution with degrees of freedom (v + ql) and correlation matrix Rl+1,1+1.(1) for 1 = 1, ... , k - 1. In the special case for R = Ir, the Y's can be written as

Yl = X1 and

y+q! Y1+1

Vv

l

+ 1v E Xm,Xm

-1/2 X1+1.

m=1

1.12 Quadratic Forms If X has the p-variate t distribution with degrees of freedom v, mean vec-

tor µ, and correlation matrix R, then XTR-1X/p has the noncentral F distribution with degrees of freedom p and v and noncentrality parameter µT R-l t/p. See Hsu (1990) for a particular case of this result. When

µ = 0, the distribution is central F and so XTR-lX/(p + XTR-lX) has the Beta(p/2, v/2) distribution. There are a number of problems related to quadratic forms of multivariate t that are worthy of further investigation.

Introduction

20

1.13 F Matrix Consider two independent random samples x11), ... , xn;) and x2),. .. , xn2)

from two different elliptical distributions (which contain multivariate t as a particular case - as already mentioned in Section 1.1). Let n;

Si

=

xW xW T k

k

k=1

for i = 1,2. Then F = (S1/n1)/(S2/n2) is the multivariate F matrix. Hayakawa (1989) studied the asymptotic behavior of the determinant, latent roots, latent vectors, and the trace of the F matrix for an elliptical population. These results are useful in the study of the robustness of the statistics derived for testing several hypotheses about parameters of a normal population with the elliptical distribution introduced as the alternative population. Hayakawa (1989) illustrated the usefulness of the results through a multivariate t-population.

1.14 Association The well known definition states that the random variables X1, ... , Xp are said to be associated if Cov(f (X1i ... , Xp), g(X1, ... , Xp))

>

0

for all nondecreasing functions f, g (Esary et al., 1967). Association im-

plies positive quadrant dependence, that is, that Pr{f1(Xi < xi)} > fp 1 Pr(Xi < xi) for all real numbers xi,... , xp (Lehmann, 1966). Jogdeo (1977) and Abdel-Hameed and Sampson (1978) established that

the components of a multivariate t random vector are associated under certain conditions on correlations. More generally, the following result holds. Let Z be a p-variate vector with independent and real components, each having a symmetric unimodal distribution. Suppose Y = Z + U, where U is independent of Z and either

(i) U = (a1V, ... , akV,

V V, .... . , anW), where (V, W) has a bivariate normal distribution centered at 0,

(ii) or U = aW, where a is an arbitrary but fixed p-variate vector and W is an arbitrary real random variable.

1.15 Entropy

21

For (n + 1) independent and identically distributed (iid) copies YT = (Yi1,... , Yip), i = 0,1, ... , n of Y define X?, j = 1, ... , p by 2 nY_j

X2 =

En

i=1

yij

Then the variables XJ2 (or, equivalently, I Xj 1), j = 1,.. . , p are associated. Now, redefine Y as a p-variate normal random vector with zero means and covariance matrix specified by E = {rid oiaj}. Let S2 and S, 2 be independent chi-squared random variables with degrees of freedom n and qk, respectively, for k = 1,. .. , p. Also assume that X, S2, and Sk2 are mutually independent. Then, as a consequence of the above general result, one could provide the following assertions about bivariate and trivariate t vectors For p = 2, the random variables

(X1, X2) =

1y11

1Y21 S2 + S22

S1 + Sit

are associated.

For p = 3, if fi<j sign(.ij) < 0, where A = {Aid} = E-1, then the random variables

(X1,X2,X3) =

IY11 Sl + Sit

1Y21 SZ + S22

1Y31

S2+S3 2

are associated.

1.15 Entropy The entropy of a continuous random vector X may be regarded as a descriptive quantity, just as the median, mode, variance, and the coefficient of skewness may be regarded as descriptive parameters. The entropy is a measure of the extent to which a multivariate distribution is concentrated on a few points or dispersed over many points. Thus, the entropy is a measure of dispersion, somewhat like the standard deviation in the univariate case. Mathematically, the entropy of X is defined by

H (X) = E [- log f (X)]

Introduction

22

= - J f (x) log f (x) dx.

(1.26)

Guerrero-Cusumano (1996a) derived the forms of this for the multivari-

ate t distribution. For a central p-variate t, it turns out that

H (X; R) =

1 log IRI + log I L

+v

(Fp/2) (2 B (2' 2) ] l v-x\21],

(1.27)

where b(t) = d log F(t) /dt denotes the digamma function. Note that (1.27) can reexpressed as H(X) = 1/2 R I +4)(v, p), where -P(v, p) is a constant that depends only on v and p. Table 1 in Guerrero-Cusumano (1996a) tabulates 4 (v, p) for v = 1(1)35 and p = 1(1)5. The following is an abridged version of the table.

Constant 4) for H(X) = 1/2 R I +F(v, p) v 1

2 3

4 5

6 7 8 9

10

p=1

p=2

p=3

p=4

p=5

2.53102 1.96028 1.77348 1.68176 1.62750 1.59172 1.56638 1.54750 1.53289 1.52126

4.83788 3.83788 3.50454 3.33788 3.23788 3.17121 3.12359 3.08788 3.06010

7.06205 5.67306

9.24381 7.48261

11.3999

5.20997 4.97687 4.83602 4.74153 4.67368 4.62257 4.58266 4.55062

6.89826 6.60362 6.42500 6.30474 6.21809 6.15261 6.10135 6.06010

3.03788

9.27502 8.57432 8.22121 8.00685 7.86226 7.75785 7.67878 7.61677 7.56678

The particular case of (1.27) for v = 1 gives the entropy for the multivariate Cauchy distribution

H(X;R) =

logR+log(2'2)] [')B p +1

2

(1 [,p

2

p)

- 7P MI

As v -* oo, (1.27) converges to the entropy of the normal distribution

1.16 Kullback-Leibler Number

23

given by (1.28) log(2e7r) + f log JR1. 2 The sampling properties of (1.27) will be discussed in Chapter 9. For the noncentral p-variate t, (1.26) takes the general form

H (X; R) =

H(X;R) = 21 logJR1+log(( (p/2)B\221]+2pM(v,p,0), (1.29)

where A = µTR-lµ and M(v, p, A) is given by

M(v

0) =

exp-

+ 2+211

00

(2)

I

j=0

Setting v = 1 in (1.29), one can obtain the entropy of the noncentral pvariate Cauchy distribution. In the case p = 1, (1.29) coincides with the entropies for the univariate Student's t and Cauchy distributions given, for example, in Lazo and Rathie (1978). Zografos (1999) provided a maximum entropy characterization of (1.1). The maximum entropy principle suggests to approximate the unknown

pdf of X by the model that maximizes (1.26) subject to the constraints that define the class of pdfs considered. Jaynes (1957) asserted that the maximum entropy distribution, obtained by this constrained maximization problem, "is the only unbiased assignment we can make; to use any other would amount to an arbitrary assumption of information which by hypothesis we do not have." Zografos (1999) showed that (1.1) is the solution to maximizing E[- log f (X)] subject to the constraint E [log { 1 + 1 (X - µ)T R-1 (X l v

/p+V; p

wf

2

2

'

)J

where w(x; a) = 0(x) - 0(x - a), x > a, and denotes the digamma function. For further discussion of maximum entropy methods, see Fry (2002).

1.16 Kullback-Leibler Number The mutual information of a continuous random vector X with joint pdf f (x) and marginal pdfs f (xi), i = 1, ... , p is defined by T (X)

= E [_log{

f(X) ... f

f (xl)

l

(xp) }]

(1.30)

24

Introduction

with the domain of variation given by 0 < T(X) < no. (The reader should not confuse this with the transformation T(X) given in (1.5).) The quantity (1.30) can be considered a measure of dependence (Joe, 1989). The larger the T(X), the higher the dependence among the variables Xi, i = 1,.. . , p. Naturally, T (X) = 0 implies that the variables are independent; this latter statement follows from the fact that T is a special case of the Kullback-Leibler number, KL(f, g) (Kullback, 1968). When the variables of X are multivariate normal with covariance matrix E, it is easy to compute T(X) as the difference between entropies given

by (1.28); specifically,

T(X;E) = H(X;E)-H(X;D), where D is a diagonal matrix corresponding to E with the elements au,... , app This is due to the well known fact that uncorrelatedness implies independence in the normal case. This fact also implies that T(X; I) = 0. In general, for any member of an elliptical family of distributions, this is not true; in other words, uncorrelatedness does not imply that T(X) = 0. The mutual information attempts to summarize in a single number the whole dependence structure of the multivariate distribution of X. Guerrero-Cusumano (1996b) derived the form of (1.30) for the multi-

variate t distribution. For a central p-variate t, it turns out that

=

T (X)

Il - 2 log JRI

(1.31)

,

where 1 is given by

Q=

(i

tog

r (p/2) BP xp/2 B (p+v a) 2

' 2P_)

+ Al + v) 2

p2v{0(p2v)-

\2/}.

{ (1

2

v)

-

(2 (1.32)

It is easy to see that fl -4 0 as v -4 no. The mutual information for the multivariate normal distribution with correlation matrix R is given by -(1/2) log I R I (Kullback, 1968). The particular case of (1.31) for v = 1 gives the mutual information for the multivariate Cauchy distribution with c taking the simpler form Il

=

log

8P r(p+2) }-12pt,,(12p)- (2)}. /2

(12)

ll JJJ

1.16 Kullback-Leibler Number

25

Table 1 in Guerrero-Cusumano (1996b) provides values of (1.32) for a range of v and p. The following is an abridged version.

Constant 52 for T(X) = 52 - (1/2) log I R I V

p=1

p=2

p=3

p=4

p=5

1

0

0.4196180

0.949615

1.530690

2.141170

2

0

0.2927000

0.705474

1.184010

1.704100

3

0

4

0

0.2254360 0.1835450

0.975130 0.832265

1.431820 1.240460

5

0 0

7

0

8

0

9

0

10

0

0.727338 0.646600 0.582368 0.529959 0.486337 0.449434

1.096790

6

0.565424 0.473177 0.407380 0.357917 0.319304 0.288289 0.262813 0.241503

0.1548760 0.1339950 0.1180970 0.1055830 0.0954730 0.0871342

0.984235 0.893344 0.818244 0.755056 0.701101

Figures 1.5 and 1.6 graph T(X) in (1.31) for p = 2 and p = 4, respectively. The correlation matrix R is taken to have the equicorrelation structure ri1 = p, i # j. It is interesting to see the "dale-shaped" threedimensional plot. The figures show that, as one moves toward the center of the "dale," the dependence among the variables decreases, and, as one moves away from the center, the dependence increases. For the normal case, Linfoot (1957) and Joe (1989) suggested a parameterization for T(X) to make it comparable to a correlation coefficient.

They defined the induced correlation coefficient based on the mutual information as PI

=

1 -exp {-2T (X)}.

(1.33)

Guerrero-Cusumano (1998) suggested a similar measure for the multivariate t distribution referred to as the dependence coefficient. It is given by

pI

=

1- I R I exp (-252).

(1.34)

The dependence coefficient is a quantification of dependence among the p variables of X. This follows from the fact that independence implies

Introduction

26

Fig. 1.5. Mutual information, (1.31), for p = 2

pi = 0 and that T(X) = oo implies pI = 1. When v - oo, (1.34) coincides with (1.33). The sampling properties of (1.31) will be discussed in Chapter 9.

1.17 Renyi Information Since the concept of Renyi information is not widely available in the literature, we provide here a brief discussion of the concept. Renyi information of order A for a continuous random variable with pdf f is defined as ZR(A)

1 1

log

(ffk(x)dx)

(1.35)

1.17 Renyi Information

27

rho

Fig. 1.6. Mutual information, (1.31), for p = 4 for A , 1. Its value for A = 1 is taken as the limit

IR(l)

lim IR(A)

A-+1/

- J f(x)log(f(x))dx

-E [log f(X)], which is the well known Shannon entropy. Renyi's (1959, 1960, 1961) generalization of the Shannon entropy allows for "different averaging of probabilities" via A. Sometimes (1.35) is also referred to as the spectrum of Renyi information. Renyi information finds its applications as a measure of complexity in areas of physics, information theory, and engineering to describe many nonlinear dynamical or chaotic systems (Kurths et al., 1995) and in statistics as certain appropriately scaled

Introduction

28

test statistics (Renyi distances or relative Renyi information) for testing hypotheses in parametric models (Morales et al., 1997). The gradient I'R(A) = dIR(A)/dA also conveys useful information. In fact, a direct

calculation based on (1.35) - assuming that the integral f f'(x)dx is well defined and differentiation operations are legitimate - shows that

(1 - a) f f 'f ZR(1) _ lim A-+1

f),\lo(g

+ log

(ffA(x)dx)]

I

f f A(x) loge f (x)dx

1 lim

f f (x)dx

2 A->i

-C

f f (x)1og f (x)dx)

2

)

f f (x)dx

-1 V ar [log f (X )J . In other words, the gradient of Renyi information at A = 1 is simply the negative half of the variance of the log-likelihood compared to the entropy as the negative of the expected log-likelihood. Thus, the variance of the log-likelihood I f := 21I(1) measures the intrinsic shape of the distribution. This can be seen by observing that If, where f (x) = (1/Q)g((x - p)/Q). In fact, according to Bickel and Lehmann (1975), it can serve as a measure of the shape of a distribution. In the case where f (x) has a finite fourth moment, it plays a similar role as a kurtosis measure in comparing the shapes of various frequently used densities and measuring the heaviness of tails, but it measures more than what kurtosis measures. Renyi information of order A for a p-variate random vector with joint pdf x is defined as IR(A)

:=

log Y f' (x1,. .. , xp) dxl ... dxp 1

1 1

.

(1.36)

The gradient IR(A) and the measure If are defined similarly. Song (2001) provided a comprehensive account of IR(A), IR(A), and If for well known univariate and multivariate distributions. For the univariate Student's t distribution with degrees of freedom v, it can be shown for A > 1/(1 + v) that IR(A)

1

-A

to

B ((vA

g

t

1)/2,1/2)

B A(v12,112)

+ 2 log(y),

1.17 Renyi Information IR(A)

-

29

{ B ((vA + A - 1)/2,1/2)

log

B (v/2,1/2)

+(1-A)2(1+v)

(VA + A-1)

,

- (1-A)(1+v) ((1+v)A)]/(1A)2 and

(1+v)2 {(v)(i+u)}

If (V)

Using tables in Abramowitz and Stegun (1965), one obtains the particular values 2 *7r

if (1)

3'

If (2)

9-

If ( 3

37x2

4

=

)

=

If (4)

47x2

3

- 12

775 36

=

1f (5)

'

37x2 -

,

257r2 12

115 4

It is interesting to note that the measure If (v) decreases as v increases, which makes sense since the tails become lighter as v increases. In fact, it can be shown, using asymptotic formulas for the trigamma function,

that

If(v) = 1/2, which corresponds to the measure If(v) for

the normal distribution. For the central p-variate t distribution with correlation matrix R and degrees of freedom v, it can be shown for A > p/ (p + v) that ZR(A)

=

1 1

log { B ((vA+

p/2),p/2) l

p A2

J

+ 2 log {(v7r) ' I R

- logy (P2) 1

log

B ((vA +pA - 1)/2,p/2) B (v/2, p/2)

+(1 - A)(p+v), (vA+A_\ 2

Introduction

30

((p +2v)\ 1

(1 - A) (p + v)

I/

2

(1 - a)2,

and

Zf(v)

_

(p4v)2

l l Ml

(±±)}.

For p = 1, these expressions reduce to those derived for the Student's t distribution.

1.18 Identities In one of the earliest papers on the subject, Dickey (1965, 1968) provided two multidimensional-integral identities involving the multivariate

t distribution. This first identity expresses a moment of a product of multivariate t densities of the form (1.1) as an integral of dimension 1 less than the number of factors. Consider the product K

g (x)

= [I

[1

+ (x - µk)T Rk (x - µk)

Uk/2

(1.37)

k=1

where each Rk > 0 and Uk > 0, and so each term may not have a finite integral. The identity seeks an expression for the complete p-dimensional integral of s g, where s(x) is a polynomial in the coordinates of x. Let Y be a p-variate normal random vector with the covariance matrix and mean vector given by -1

K

Du 1

=

(>ukRk) k=1

and

µu = D'

K

ukRkµk1 k=1

respectively.

Ek 1 CkUk

For given constants Ck > 0, k = 1, ... , K, let u. = and uk = Vku.. Then the quantity defined by N8lu =

E(s(Y)) can be expanded as a polynomial in 1/u. as N8Iu

= E hj(v1i ... , vK) u7

1.18 Identities

31

Given this terminology, the identity can now be expressed as

s(x)g(x)dx fn P

2-jr

KO

X

(V.2 p

ID,I-1/2h.7 (v1i...,VK)

-j/ J

K

Wv-(- -p)12dv1 ... dvK-1,

K

(1.38)

k=1

where K

Ko =

v/2 7r

fl r (vk/2), k=1

V.

K _ E vk, k=1

K

Dv =

k=1

vkRk,

1

K

WV

= E Vk

K

T

K

f, + /lk Rkltk llI -

vkRkµk

Dv 1

vkRkILk

(k=1

k=1

k=1

and o, is the simplex K

1

Q = {(v1. .. , 4JK) : 1: Ckvk = 1,

vk> ()

k=1

This identity has applications to inference concerning the location parameters of a multivariate normal distribution. In the particular case K = 2, Rk = ryklp, and s =- 1, (1.38) reduces to

f

g (x) dx =

Cy2U.-p)12

B

Rp

Vi

xF1 (

v v2

\2

2

v. - p v. - p v.

2

2

'

2

where C

xpl2

IF ((v. - p) /2)

r(v1/2) r (v2/2)

vi/2 v2/2, 71

'Y2

2,

z1, z2

,

(1.39)

Introduction

32

F1 is Appell's hypergeometric function of two variables defined by F1 (a;

_ _

Q'; 7; x, y) 1

1

B(a,7-a)

to-1(1

-

t)7-a-1(1

-

tx)-a(1

- ty)-R'dt

o

(1.40)

(see, for example, Erdelyi et al., 1953), and z1 and z2 are the two real roots of the equation

z2+(7211 µ2-µl112+ryi-llz-7211 µ2-µ11I2 =

0.

The integral (1.39) is proportional to a multivariate generalization of the Behren-Fisher density. For an asymptotic expansion of (1.37) in powers of vk, see Dickey (1967a). The second identity given by Dickey (1968) - see also Dickey (1966b)

- expresses the density of a linear combination of independently distributed multivariate t vectors as an integral of dimension 1 less than the number of summands. Consider the r-variate vector 6 formed by the linear combination K

= E BkXk,

6

k=1

where Xk are independent qk-variate standard t random vectors with zero means, covariance matrix I9k , and degrees of freedom vk. Dickey (1968) showed that b has the representation S

=

vkU; 1BkBkY, k=1

where Uk are independent chi-squared random variables with degrees of freedom vk and Y is an independent r-variate standard normal vector. As a consequence, 6 has the further representation K k=1

vk, Vk = Uk/ >K1 U, and W is an independent where v. = 1 r-variate standard t vector with degrees of freedom v.. If the matrix E BkBk is nonsingular, the distribution of 6 is nondegenerate with the

1.19 Some Special Cases

33

joint pdf K

=C

f (S)

I

-(v.+r)/2

K

ft

1 + ST

(vk/vk) BkBk

k=1

k=1

K

(vk/vk) BkBT dv1 ... dvK-1,

(1.41)

k=1

where

r ((v. + r) /2)

C = 7.r/2r (v1/ 2) ... r (vK/2) and as above K or

=

{(vi. ..,VK) : Y Ckvk = 1,

Vk > 0

k=1

This identity has applications in Behrens-Fisher problems. The version

of(1.41)forK=2andBk=/3kis f (S)

= CB

(v1 + p v2 + p1 \\

2

xF1

J

2

'

v1+p v.+p v.+p V. 2

2

,

,

2

, 2 +A Z1, z2

where

C-

r((v. +p)/2) 7r n/2r (v1/2) ... r (v2/2)

(v202)-(U1+n)/2,

(v1Q2)vl/2

F1 is Appell's hypergeometric function as defined in (1.40), and z1 and z2 are the two real roots of the /equatioon z2 + r II S I2 + v2#22

V, 012 V2 #22

1Iz.1

II S II2

=

0.

1'2182

This special case is essentially equivalent to the two-factor version of (1.38). Moreover, (1.41) is a generalization of Ruben's (1960) integral representation (in the univariate case) for the usual Behrens-Fisher densities.

1.19 Some Special Cases A number of special cases of (1.1) have been studied in the literature with great detail. Cornish (1954), in his early paper, considered the

Introduction

34

special case of (1.1) when µ = 0 and R is given by the equicorrelation matrix 1

-1/p

R =

-1/p ... -1/p ... -1/p 1

-1/p -1/p ...

1

/

The following interesting properties were established

XTR-1X has the noncentral F distribution with degrees of freedom p and v. XTR-1 X has the Fisher's z distribution with degrees of freedom p - q and v - when X is subject to the linearly independent homogeneous conditions represented by the equation SX = 0, where S is of order

gxpandrank q
r ((v + q)/2)

(7rv)q/2r (v/2) f ...

°+q)%2

xTxl1

C1 +

v

Y

dxl ,

where xT = (x1, ... , xq) and the domain of integration is defined by q

fix? > Q, i=1

where Ai are the roots of the equation I AR-1 - A I= 0 or, alternatively, the latent roots of the matrix RA. Consequently, the distribution of XT AX is Fisher's z with degrees of freedom q and v if and only if the nonzero latent roots of RA are all equal to unity. If the distribution of X is partitioned as in (1.11)-(1.13), then

E (X1 I X2) = -R111R22x2, and

Var (X1 I X2) =

V + x2 (R22 - R21R111R12) x2

v+p-pl-2

In the particular case pi = 1,

r

E(X1 I X2) _

2T xj, j=2

-1

1.19 Some Special Cases Var (X1 I X2)

_

(p+1)v

35 3

p

2p(v + p - 3) + 4(v + p - 3) j=2 =2

p+1

+2(v+p-3) E[Var(X1 I X2)] = Var (X2)

=

EX'xk'

v p+1

v2 2p v

v 2822 V

i = 2, ... , p. - p(v - 2)' Furthermore, the residual variance of Xl with respect to X2 is Cov (X1i Xi)

v p+l

v-2 2p

'

and the partial correlation coefficient of X1 with respect to X2 is

-1/2. Patil and Kovner (1968) provided a detailed study of the trivariate t density

f (x1,x2,x3)

F((n + 3)/2) (nr')3/2 1 - p2r (n/2) X

Cl +

-(n+3)I2 1x1 - 2px1 + x22+ x223 I n 1-p2

Among other results, Taylor series expansions - in powers of 1/n - of the density and associated probabilities in rectangles were given.

2

The Characteristic Function

The characteristic function (cf) of the univariate Student's t distribution for odd degrees of freedom was derived by Fisher and Healy (1956). Ifram (1970) gave a general result for all degrees of freedom, but Pestana

(1977) pointed out that this result is not quite correct. More recent derivations are presented in Drier and Kotz (2002). Here we discuss two independent results on the characteristic function for the multivariate t distribution. The first one, due to Sutradhar (1986, 1988a), provides

a series representation for the cf while the other, due to Joarder and Ali (1996), derives an expression in terms of the MacDonald function. The expressions given are rather complicated; thus, further research and possible simplifications may be desirable.

2.1 Sutradhar's Approach Let X be distributed according to

f (x)

=

F'1 v +p)12)

(irv)P/ r (v12) IRI 1/2

1 + 1 (x - µ)T R (x - µ)1

-(,+n)/2

J

(2.1)

Consider the transformation Y = R-1/2(X - µ). It then follows that the joint pdf of Y is

=

vv/212) r(v12)

36

(v I

\

P

+

E Jk

k=1

(v+n)l2

2.1 Sutradhar's Approach P

F ((v +p)/2)

exp (itT y)

v-v/2r (v12)

37 (v+P)/2

v+Ey

dy1 dye ... dyP

k=1

To evaluate this integral, Sutradhar (1986) makes the orthogonal transformation Y = I'Z, where the first column of the p x p matrix r is the vector t2

t1

tp

IltIIIItIII

'Iltll

tTt. It follows that the cf of Z is given by

with 11 t 11=

Oz (t; v)

/2 F

=

vV

x

v ((r (

p) 2) f exp (i II t II z1) dzi

f f ... f (cp + z2)-(v+p)/2 dz2 ... dzp, (2.2)

where zk E R, k = 1,. .. , p and cp = v+ EP-1 zk. Successive integration of (2.2) with respect to zp, z _ 1 , . .. , z2 yields

=

4z (t; v)

L,/2(( v + 1)/2) J1 vPr (v/2)

(2.3)

where 00

J1

=J

-(v+1)12

exp (z II t ll w) (+u2 + v)

dz.

(2.4)

00

Note that J1 is an improper integral along the real axis, where w denotes a complex variable. For odd v, the integrand has poles of integer order

while, for fractional and even v, the poles are of fractional order. Sutradhar (1986) evaluated J1 separately for the three cases: odd v; even v; and fractional v, using the relations that

= exp (itT µ) OY R' /2t- v

Ox (t; v)

(2.5)

and

OY (t; v)

= Oz (t; v)

(2.6)

to obtain the following expressions for Ox. For the case of odd v V17r

x(t; v) =

F ((v + 1)/2) exp (itT, 2"-1r (v/2)

X k=1

-

vtTRt)

(2. - k - 1 (2 vtT Rt) (k-1)! Tn, - k

k-1

The Characteristic Function

38

where m = (v + 1)/2. For the case of even v (-1)m+1I'

_

OX (t; v)

((v + 1)/2) exp (itT ft)

V,7rI' (v/2) r', (n - k + 1/2) x

00

vtT Rt

1

iz n-0 (n) +

2n

2

f1(n - j) (log vtTRt 4 j-0

m-1

m-1

j=O

k=o>koj

11

(n - k)

- 1"(n + 1) I'(n+1))

(2.7) '

where m = v/2. Finally, when v is of fractional order OX (t;

v)

-

7r(-1)mv(v/z)-mr ((v + 1)/2) exp (itTµ)

1/2) r' ( - k) zn vtTRt 21uko1(n--k)

2T (v/2)

1

1

X

n-0

n

- (tTRt)

Uer(n + 1 -

2

!

IT- of (n - k)

(2.8)

where m = [(v + 1)/2] is the integer part of (v + 1)/2 and _ (v/2) - m is such that 0
2.2 Joarder and Ali's Approach An integral representation of the MacDonald function is

Ka(r)

r(a+1) 1

=

r (2)a

v/7

(1+u2)-(a+1/z)cos(ru)du,

(2.9)

o

where r > 0 and a > -1/2 (see, for example, Watson, 1958, page 172). A series representation of Ka (r) for r > 0 and a > 0 is given by K. (r)

(-1)j (aj' j - 1)!r2j-a

= 2a-1 j=o

.4 +(-1)a2-(1+a)

b(1 +A r2j+a j!(j + a)4j j=o Oc'

2.2 Joarder and Ali's Approach

-(-1)a2-(1+a)

39

0(1 + a + j) 2j+a r

00

j!(j + a)4j .

-(-1)a2-(1+a)

E

log (r2/4) r2j+a

00

(2.10)

j!(j + a)4

.

(see, for example, Lebedev, 1965, page 107, 110). A series representation

of Ka(r) for r > 0 and nonintegral positive values of a is

r2j-a

°°

°°

Ka(r) = 2`1 j=0 j!(1 - a)j4j

+ 2-(1+0')

1`

r2j+a

j=0 j!(1 + a)j4i

(2.11)

(see, for example, Spainer and Oldham, 1987, Chapter 51), where (c)j = c(c+1) . . . (c+j-1) denotes the ascending factorial. Using (2.9), Joarder and Ali (1996) rewrote the integral (2.4) as J1

f00

=

{cos (II t II w) + i sin (II t II w)} ao

2v-(v+1)/2 f

/

CIO

cos (II t II W)

w22

v

2

dz

1+1 w2)-

2

C1 +

v

\

o

\C1 +

dz

F(11 t

2 v/z

Kv/z (tip II t

(v + I

II)

.

Thus, using (2.3), one obtains 11

Oz (t)

=

2u

/2 r(v12)

(II ,/it

Hence, using the relationships (2.5) and (2.6), one arrives at the expression for the cf given by Joarder and Ali (1996)

Ox(t)

= exp (itT µ) 2I v/2vRt('I 2) Kv/z (II

t II)

(2.12)

Joarder and Ali also provided expansions of this cf using the series representations (2.10) and (2.11). For positive and even v, applying (2.10), one obtains v/2-1

ox (t)

= exp (itT µ)

Ci (j) II

vRt I I2j

j=0 co

+

1: C2 (j) 11 IVRt III+2j j=0

The Characteristic Function

40

00

- E C3 (j) II

vRt IIv+2j log (II v t II)

j=0

where

Ci(j) = C2(j)

-

(-1)1 (v/2 - j - 1)! (v/2 - 1)!j!4j

0(1+j)+0(1+v/2+j)+log4 2v (v/2 - 1)!j! (v/2 + j!) 4i

and 21-v4-i

(v/2 - 1)! (v/2 + j)! For positive and odd or fractional v, applying (2.11), the cf (2.12) becomes 00

Ox (t)

exp (itTµ) j=0

{Di(j) IIt II2j +D2(j) II

vRt IIv+2j}

,

where 4-j

Di(j) =

j!(1 - v/2)j

and 2-v4-3F (-v/2)

D2(j)

F (v/2) (1 + v/2) j

Since the univariate Student's t, multivariate Cauchy, and Pearson's type VII are all particular cases of (2.1), the corresponding cfs in terms of the MacDonald function can be obtained from (2.12). They are as follows

For the univariate Student's t distribution with the pdf jr

v7rF (v/2)

(1+v)/2

1 + x2

F ((1 + y)/2)

(

v

)

(where x E IR and v > 0), the cf in terms of the MacDonald function is

t Ox (t)

=

2v/2

41II, (v/2) Kv/2

(compare with Dreier and Kotz, 2002).

(v'V I t

2.3 Levy Representation

41

For the p-variate Cauchy distribution with the joint pdf

r ((1 +p)/2)

(x)(1+p)/2 IR,I1/2 [1 + (x

f (x)

- µ)T R-1 (x - µ)] - (1+p)/2

(where x E WP), the cf is Ox (t)

= exp (itTµ)

2/;r II vt II1/2 K1/2 (II vt II)

= exp (it T p- II Aft II) it/(2r) exp(-r)

which follows by using the result that K1/2(r) = (see, for example, Tranter, 1968, page 19).

For the p-variate Pearson type VII distribution with the joint pdf

f (x)

=

r(m)

xp/2r (m - p12) IRI 1/2

11

+

(x - µ)T

R-1 (x

- µ)]

_m

v (where x E RP, m > p/2 and v > 0), the cf becomes

II t IIm-p/2 = exp (itTµ) 2m,-p/2-1r (m - p/2) Km-p/2 v

Ox (t)

II

vRt II

.

2.3 Levy Representation Infinite divisibility of the univariate Student's t distribution was first proved by Grosswald (1976) - see also Kelker (1971) for a partial result. Later, Halgreen (1979) established the Levy representation of its cf. For a multivariate t, Takano (1994) provided the first proof of infinite divisibility and the corresponding Levy representation. Consider the standard

case µ = 0 and R = Ip. In this case, after suitable transformation, the joint pdf (2.1) can be written in the form

r (m r (/))

f (x) =

(1+ II x

112)-(m+p/2)

The corresponding cf is

21-r` 0 (t)

=

r(m) IItIImKm (IItII)

.

(2 . 13)

Takano (1994) derived the Levy representation of (2.13) in the form (t)

= exp f.,

lexp (itTx) -1II X lip

itTX 1+ 11 X II2

The Characteristic Function

42

x U000 gm(2w)Lp/2 ( 2w II X 11) dw} dxl ... dxr where x

La(x) =

(2ir)-axaKa(x),

f2aI (a + 1/2)

J. (x)

'2

1

z2)a-1/2

(1 -

exp (ixz) dz,

and

=

Ya(x)

1

sin (a7r)

{cos (air) Ja(x) - J-a(x)}

.

Note that JJ(.) and

are the Bessel functions of the first and second kinds, respectively, of order in. Now consider (2.13) itself as a joint pdf fm (x)

(2.14)

= C 11 x 11m K. (11 X Il)

where the normalizing constant C is 21-m-p/2 r, = (2ir)p/21' (m + p/2)' Using properties of the MacDonald function Ka(), (2.14) can be reduced to the simpler forms f1/2 (X)

= n

fn+1/2 (X)

2eXp(-IIx2kkn(n )k)

=

11Xlln-k'

k=0

and

n-I f. (X)

=

C2'"-1

E

((Akn

- 1)!

11

X

112k

k=0 00

1

r11X IIl

k!(n+k)! (\

2(n-1-k)

2

x{log("2 -)-1,P(1+k)-' (1+n+k)}.

2.3 Levy Representation

43

Takano (1994) established further that the joint pdf (2.14) is also infinitely divisible and that its cf (t)

_ (1+ II t

II)-(m+p/2)

admits the Levy representation fi (t)

= exp

[(2m + p)

f {exp (itTx) P

where Lp/20 is as defined above.

-1}

Lp/2 (11 11)

11 x lip

dxl

.

dxp]

3

Linear Combinations, Products, and Ratios

3.1 Linear Combinations The distribution of linear combinations of independent t random variables has been studied by numerous authors, among them Fisher (1935), Chapman (1950), Fisher and Healy (1956), Ruben (1960), Patil (1965), Ghosh (1975), and Walker and Saw (1978). Johnson and Weerahandi (1988) tackled the distribution of linear combinations for multivariate t random vectors. Their results are included here for completeness and to motivate further multivariate extensions. We hope that our readers will benefit from studying this material, which contains fruitful ideas and also refers to the original papers for further details. Chapman (1950) considered the difference D = X1 - X2 of two independent t random variables Xj with common degree of freedom v. If v is odd, then it is known that the characteristic function of Yj = Xj// is

0,(t) = E [exp (itYj)]

/exp (- I t I)

(v-1)/2

((v - 1)/2 + k)! 12t 1-2 ' -k v!((v - 1)/2 - k)! V

2s-1F(v/2)

k=O

(3.1)

Using this representation, Chapman provided the following general ex-

pression for the pdf of W = D/'/ f 00

f (w)

=

2J

exp {-i(v + 1)t} 02(t)dt, V 00

which may be integrated to obtain the pdf of D in a closed form for values such as v = 1, 3, 5, and so on. Chapman tabulated the distribution of D

for v = 1, 3,5,7,9, 11. 44

3.1 Linear Combinations

45

Fisher and Healy (1956) considered the mixture D = a1X1 + a2X2 of two independent t random variables Xj with degrees of freedom vj when aj > 0, j = 1, 2. It is obvious that the characteristic function of D is the product

c,, (al It 1)0v2(a2It1), is as defined in (3.1). Since the product is simply a polynomial in t of degree (vl + 112)/2 - 1, it can be expanded in a finite series of terms of in which the highest value of m = vl + 112 - 1. For example, in the special case 111 = 3 and 112 = 5, one can write where

exp {(sin 9 + cos 9) t} 03 (sin Bt) 05 (cos Bt)

25/tang (V/-5

+

+

7sin9+

-,,f3- tan 0) 3

5v/5- -10ytan0+9ftan2B

(V+ 'tan0)3

7cos9 t [(V/Jsin8+ 05

cos9)t,

3

+

(_'Jtan9

03

[(V3- sin9+ti5- cos9)tJ;

from this one can easily deduce the pdf of D. Ruben (1960) provided results on the distribution of D = X1 sin 0 X2 cos 0, when Xj are independent t random variables with degrees of freedom vj and 0 is a fixed angle between 0 and 90 degrees. This statistic was originally proposed by Fisher (1935) as the basis for testing or estimating the difference in means of two unconnected and totally unknown normal populations, the "fiducial distribution" of the difference between the latter quantity and the corresponding sample mean difference, when suitably standardized, being supposed to be that of the statistic D. Ruben obtained the pdf of D in the integral form

rl

f(d) =

f

{

1

vl + v2B((vl + 112)/2,1/2)

d202(S) }_(V1+V2+1)/2

1+

111 + 112

- s)(v2/2)-1 ds, x(s) S(vl/2)-1(1 B (vl/2,112/2)

where

0(s)

=

(U1 + 112)8(1 - s)

u1(1 -s)sin20+v2scos29

(3.2)

Linear Combinations, Products, and Ratios

46

It follows directly from (3.2) that D may be expressed in the form D

X

=

0(S)'

where X is a Student's t random variable with degrees of freedom v1 +

v2 and S is a Beta variable with parameters v1/2 and v2/2, that is, a variable with pdf given by the second term under the integral sign in (3.2), with the first term under the integral sign representing the conditional pdf of X/O(S) for fixed S. In the special case vl = V2 = v and 6 = 45 degrees, (3.2) reduces to f(d)

- ev

F(v + 1/2)F((v + 1)/2)

1

F2(v/2)F(v/2 + 1)

2

2

d2

v + 1. v +

(v +

Fl

2

2

I;-

2v

where 2F1 denotes the Gauss hypergeometric function. By using the appropriate four of the group of 24 transformations of the hypergeometric function 2F1 (see, for example, Whittaker and Watson, 1952, page 284), the above pdf may be expressed in the following three additional ways

f(d)

2 F(v + 1/2)F((v + 1)/2) V

F2(v/2)F(v/2 + 1) x 2F1 (

f (d)

_

2

(i+) 2v1-v d2

1 v+2 d2 '2' 2 ' 2v)

2 F(y + 1/2)F((y + 1)/2)

V-v r2(v/2)r(v/2 + 1)

(v+1/2)

d2

(1 + 2v)

1 v+2 (v+,;;d22), d2

1

X 2F1

(3.3)

'

and

f (d)

=

(u+1)/2 /-2F(v + 1/2)F((v + 1)/2) d2 F2(v/2)F(v/2 + 1) (1 + 2v) VV

x 2F1

(

l+v 1-v v+2 2

'

2

'

2

d2

' d2 +

2v) (3.5)

Note that (3.3) and (3.5) may each be expanded as a terminating series (refer to the definition of the Gauss hypergeometric function) when v is odd, and also that (3.4) is expressed as the product of the pdf of a t random variable with degrees of freedom 2v and the Gauss hypergeometric function.

3.1 Linear Combinations

47

In the special case v1 = no and v2 = v, (3.2) reduces to

f(d)

;v+2' 2 sin

F(v)F((v + 1)/2) (v + 1 IF, 2 27r sin OF(v/2)F(v + 1/2)

=

1

d2 z

o

where 1 F1 is the confluent hypergeometric function. Using Kummer's

first transformation for the confluent hypergeometric function, one can obtain the alternative form d2 d F(v)F((v + 1)/2)

fO

-

27r sin BF(v/2)F(v + 1/2) 1

V

x 1F1

(2;v+

exp

C- 2 sinz 6

d2

2; 2sin2 9)

Ruben (1960) also provided expressions for the cdf of D, but these were infinite series involving incomplete gamma function ratios. For tables of percentage points of D, see Sukhatme (1938), Fisher and Yates (1943), and Weir (1966). Ghosh (1975) provided explicit expressions for the cdf in terms of simple hypergeometric functions when D = X1-X2 and X3 are independent Student's t random variables with common degree of freedom v < 4. In particular, Ghosh obtained the following expressions for Pr(0 < D < d) 7r

1

arctan (d) 2

{(1 + q)E(q) - (1 - q)K(q)} ,

2/d (18 + dz) 7r (12 + d2)2

1

+

arctan

(d1

and

1 (8p4 -3 1P3 + 48p2 + 5p + 2) E(p)

64p

- (4p4 - 16p3 + 6p2 + 4p + 2) K(p)

,

for v = 1, v = 2, v = 3, and v = 4, respectively, where p = d2/(16+d2),

q=d2/(8+d2), n/ z

E(x) =

1 - x sinz sds fo

48

Linear Combinations, Products, and Ratios

is the complete elliptical integral of the second kind, and 7r/2

K(x) =

ds

1-xsin2s

Jo

is the complete elliptical integral of the first kind. Similar expressions

for Pr(0 < D < d) as a finite sum of terms can be obtained for any positive integer v. Ghosh also provided a tabulation of the numerical values of Pr(D < d) for v = 1(1)20 and d = 0.0(0.5)10.0. Walker and Saw (1978) expressed the cdf of the linear combinations of any number of Student's t random variables with odd degrees of freedom as a mixture of t distributions. Define the linear combination as n

D=E

ak

vkXk,

k=1

+ an = 1 and Xk are independent Student's t

where ak > 0, a1 +

random variables with degrees of freedom vk = 2mk+1, Mk = 0, 1, 2, .... Construct a matrix Q whose element in row i and column j is the coefficient of exp(- I t I) t Ii in 02i+1(t) (see equation (3.1)), that is, I

2ji! 2i-j ! (2z)I'Iz j)

I

0

'f -0 1 2 if j > i.

-0 1 2

The characteristic function of D when vk = 2mk + 1 can be written as n

fi(t) = E [exp (itD)] = 1 10,, (akt)

,

k=1

and since exp(I t I)i(t) is a polynomial in I t I, one may obtain a vector A such that fi(t)

= exp(- ItI)

AkItIk. k-o

Walker and Saw (1978) showed that the cdf of D can expressed as the weighted sum

s Pr(D < d) =

E%Hk(d),

k=0

where 77

T = xTQ-1

,

3.1 Linear Combinations

49

n

S=

Mk, k=1

and

= Pr [X2k+1 <

Hk (d)

2k + ld]

.

This result can be used to calculate probabilities of D utiliizing only tables of the Student's t distribution. The general distribution of D = X1 - X2 is very complicated when the Xj are independent Student's t random variables with v1 # v2. It is therefore natural to ask whether a reasonably good approximation can be found. Chapman (1950) suggested the simple approximation

dl

Pr (D < d)

y2

y1

,122

+ v222

,

where 4 is the cdf of the standard normal distribution. This idea is, of course, prompted by the fact that X1 vl - 2/ v1 and X2 Vv-2- 2/ v2 are both asymptotically standard normal random variables, approaching normality more rapidly than X1 and X2 do. However, a few calculations show that this approximation is quite unsatisfactory even for moderately

large values of v1 > v2 > 2. Based on a t-approximation proposed by Patil (1965), an improved approximation is

Tv O

Pr (D < d)

where T is the cdf of the Student's t distribution of v degrees of freedom,

v = 4+

cost y2 + sing V, vi-2 V2-2

2

SIn4 V1

COSQ i/2

(vz-2)

(v

(where vl > v2 > 4), and h2

v

v-2

cost v2

(v2-2

Sin2 U

+

v1-2)'

where v1 > v2 > 4. Ghosh (1975) derived another approximation that requires only tables of the normal distribution Pr(D < d)

-

(D

(d) - dO(d/y) IQ, (d) + Q2 (d) + 32i

V2

v2

Q3 (d) v2

Linear Combinations, Products, and Ratios

50

where

= (1+A)(d2+10),

Q1(d)

=

Q2 (d)

1 + A2 384

(3d6 + 98d4 + 620d2 + 168)

+64 (d6 - 1044 + 3642 -456),

Qs (d)

=

1+

A2

24576

(d10 + 66d8 + 1016d6

+A(1 + A) (3d10 24576

- 1296d4 - 65328d2 -141408)

- 58d8 - 280d6 + 6864d4 - 70032d2)

1277A(1 + A)

+ 256 and A = v2/vl. Ghosh showed evidence to suggest that this is far more accurate than Patil's approximation. Johnson and Weerahandi (1988) considered linear combinations of t random vectors in a Bayesian context. Suppose Yii,... , Y1,m1 and Y21, . , Y2M2 are independent samples from two p-variate normal populations N(µ1, E1) and N(µ2, E2), respectively, where the population covariance matrices are unknown and not necessarily equal. Let Yi and Si denote the corresponding sample mean vectors and sample covariance matrices. Johnson and Weerahandi considered the distribution of the quadratic form

Q=

(6 - d)T

V-1

(S - d)

,

where S = Al - µ2, d = Yl - Y2, and V is any p x p positive definite matrix. Note that µi - Yi have the central p-variate t distribution with covariance matrix Si/(mi - p) and degrees of freedom mi - p. Under the diffuse prior distribution

p(µ1,El,µ2) E2) _

IE1EiI,

the posterior cdf of Q can be expressed as 00

F(q)

nq

_ E E (wj) Fp+2j,n 10(p+2j)] ,

(3.7)

j=o

where n = m1 +m2 - 2p and the expectation is taken with respect to the

3.1 Linear Combinations

51

beta random variable B, which is distributed as Be((ml - p)/2, (m2 p)/2). The wj are defined in terms of 0 (an arbitrary constant) and )j by the recursive relation 1 r-1

Wr = 2r E Hr-jwj, j=0

where wo

Hr

j Ir

=

e

j=1

and Aj are the ordered eigenvalues Al < ... < A of the matrix

m1BV-1/2S1V-1/2 + 1122(11V-1/2S2V-1/2. B) In the particular case V = cS1 + (1 - c)S2, the Aj can be conveniently obtained by using the relation

m2(1 - B) + m1m2B(1 - B) {c+ (1 where 1, ... , p are the eigenvalues of ST 1S2. In the univariate case,

(3.7) reduces to give the posterior cdf of Y = (Al - µ2) - (-ti - x2) as

-m2 -2 -1i+ =1 1-B (_ymi ml B M2 mz where s1 and s2 are the sample variances and the expectation is taken with respect to B, which is distributed as Be((ml - 1)/2, (m1 - 1)/2).

The result (3.7) can also be used to deduce the pdf of U = (T1 ± T2)T(T1 ± T2), where the Ti are independent p-variate random vectors having the t distribution with covariance matrix (ai/mi)Ip and degrees of freedom mi. It turns out that the cdf of U is

F(u)

r

= E ILFP,ml+mz

u(ml+m2) p

B(1-B) a1 + B (a2 - ai)

l

where the expectation is taken with respect to B, which is distributed

Linear Combinations, Products, and Ratios

52

as Be(mi/2,m2/2). Johnson and Weerahandi also established several interesting bounds on (3.7), one upper bound being F(q)

< E Fp,_

p m1BS1 + m2(11- B)S2

V-1/2SiV-1/2. Furthermore, it was shown that similar rewhere Si = sults hold if the diffuse prior in (3.6) is replaced by the natural conjugate prior distribution.

3.2 Products The distribution of products of Student's t random variables has been studied by Harter (1951) and Wallgren (1980). Products of Student's t random variables arise naturally in classification problems. In many educational and industrial problems it may be necessary to classify persons or objects into one of two categories - those fit and those unfit for a particular purpose. In formulating a classification problem, assume that for p tests one knows the scores of N1 individuals known to belong to population 11, and of N2 individuals known to belong to population II2, along with those of the individual under consideration, a member of the population H, where it is known a priori that 11 is identical with either I'll or 112. Assume further that the distribution of the test scores of the individuals making up 11, and 112 are two p-dimensional normal distributions, which possess the same covariance matrix but are independent of each other. In order to classify the individual in question into either 11, or 112i Wald (1944) introduced the statistic V given by P

P

V =

ij S Yi,n+1Yj,n+2,

i=1 j=1

where n = N1 + N2 - 2, sij is the (i, j)th element of the inverse of the matrix S defined by 1

sij

n

= n E Yi,kYj,k, k=1

and Yi,k (i = 1 , ... , p; k = 1, ... , n + 2) are iid normally distributed random variables with unit variances and expected values E(Yi,k) = 0, k = 1,... , n, E(Yi,n+1) = µi, and E(Yi,n+2) = µ2

3.2 Products

53

In the particular case p = 1, (3.8) can be written as

V=

Yi,n+l_i,n+2 Z

where

Z =

Yz

2

k=1

Thus, one sees that the V in (3.9) is a product of two independent Student's t variates. Harter (1951) derived the exact distribution of this V. In particular, he showed that the pdf of I V I is given by 00

1

f (I V I)

_ err (n/2) xr2

n/2+j

j=O

1:(-1) n

Iv

(1+j+n/2)

(2+2i+n)/r(li)

ifIvl>n/2, and by f (I V I)

=

exp (-µ2/2) 00

xr2

IV

r(1 + j)

7rf' (n/2)

I-(1 {+n/2)

(2+2j+n) - (2µ2)kr (k+ 2+n+2jl 4

/f k (2k)!

4

J

ifIvl
ates. This was motivated by hypotheses testing problems that assume that the relationship between two regression lines y = a1 + /31x and y = a2 + /32x hold for all real x. Although it is often true that such a relationship holds for all real x, there are instances in which the relationship may hold true only for x's in a given interval, say [c1, c21. Wallgren (1980) showed that the statistic for testing this hypothesis is a product of two correlated t-variates; specifically, it takes the form W = XY/S2, where (X, Y) has a bivariate normal distribution with means (Al, p2), common variance Q2, and the correlation matrix

R

( 1

1

1 .

(3.10)

Moreover, nS2/U2 is independent of (X, Y) and has a chi-squared distribution with degrees of freedom v.

Linear Combinations, Products, and Ratios

54

The limiting distribution of W = XY/S2 as v -4 oo is Z = XY/Q2. The distribution of Z has been studied by Aroian et al. (1978). The study of the distribution of W is, therefore, a generalization of the study by Aroian et al. since Q2 is unknown. . In the central case pi = 0 and µ2 = 0, Wallgren (1980) showed that the cdf F(w) = Pr(W < w) of W is given by

1-

F(w)

fQ,(B;p,w)dO 0

for-10; F(w) _

0

a-n

Q (9; p, w) d9

for-1
forl>p>0,w>0;and F(w) _

1a OQv(B;P,w)dO

for 1 > p > 0, w < 0, where

irQv (9; P, w)

v sin 0 sin (0 + A)

=

( w + v sin 0 sin (0 + A) )

a = arctan -

v/2 7

1 - P2

P

and angle A is defined by sin A = 1 - p2, cos A = p for 0 < A < ir. The corresponding pdf f (w) can be obtained by differentiation. The pdf has a singularity at w = 0 and, considered as a function of p and w, f (w; p) = f (-w; -p). The limit of f (w) as p -* 1- is the F1,,, density. Moreover, if pl < P2, then F(w; pi) > F(w; p2) for any w. In the noncentral case, the cdf F(w) is given by

F

=

f f o

J

h(s) exp a2

(-v2/2) 27r

1 x' (WS 2/(v+A2)-Al-PV dvds 1-p2 J

3.2 Products

55

or, equivalently, 00

F

f -A2 h(s) exp (-y2/2)

J

o

27f

o0

+ A2) + A1 +pvl

J

X (-ws2/(v 1-p2

duds

where \i = µi/a, i = 1, 2 are the noncentrality parameters and vv/2sv-1 exp (-vs2/2) 2(v-2)/2F (v/2)

The two double integrals above are bounded above by 4)(A2) and 4)(-A2), respectively. As a function of Al and A2, F has the following properties:

F(w;A1,A2) = F(w;A2,A1), F (w; -Ai, -A2) = F (w; A1, A2)

F(w;a1,-A2) = F(w;-A1,A2), lim F (w; cA2, A2)

= 0 if c > 0,

A2--*00

lim F (w; cA2i A2)

A2- 00

lim F (w; cA2i A2) A2-*00

=

1/2 if c = 0,

= 1 if c < 0,

lim F (w; A1, B) _ 4)(-B),

A1-*00

and

lim F (w; a1i B) =

ai-r-00

(P(B).

Also, for Al > 0, A2 > 0, w > 0, and -1 < p < 0, F(w; A1, A2) is a strictly decreasing function of Al and A2. Thus, the maximum of F(w) over the region Al > 0, A2 > 0 occurs at Al = A2 = 0. Since (X, Y) and S2 are independent, the first two moments of W are given by

E(W) = E (XY) E (1/S2) and

E (W2)

= E (X2Y2) E (1/S4)

.

Linear Combinations, Products, and Ratios

56

It is known (see, for example, Craig, 1936) that E (XY) = Q2 (A1.A2 + P)

,

2A2 E (X2Y2) = Q4 (A2 +1 A2 + 2 4pAIA2 + A 1 2

+1+2 P2 )

,

and

-

E

v,r (y/2 - i)

L(1/S2)iJ

(2a2)Z P (v/2)

forv>2'. 3.3 Ratios

For a bivariate t random vector (X1, X2) with degrees of freedom v, the mean vector (mi, my) and correlation matrix I2 define the ratio W = X1/X2. The distribution of this ratio is of interest in problems in econometrics and ranking and selection. Press (1969) derived the pdf of W as

f(w) = 1+w2

+

k2q

1

{2T Cq

V

q*

+1

I -1}

-oo<w
(3.11)

where denotes the cdf of the univariate Student's t distribution with degrees of freedom v + 1 and k1i k2, q, and q* are constants defined by _v/2 + + my kl mX2

=-

v

7r

V 7rv(v+2)/21+ ((v + 1)/2) k2

1 + m2 + my

-v/2

v

2F ((v + 2)/2)

mXw + my q

--

1+w2

and q*

=

m2 + TO + v - q2,

respectively. In the special case mx = my = 0, (3.11) is the pdf of a Cauchy distribution. The asymptotic distribution of W as v oo is

3.3 Ratios

57

that of the ratio of independent normal random variables. This fact can be verified as follows. Let f,,,, (w) = f(w). Then it is easy to check that lim

lim

_

k2

1

20(q)

=

V1rV_ _+1

V 400

q*

lim

1,

4)(x),

v-roo and

lim kl

=

1 exp { Ti.

111

-1 (m + my) 2

111

where ¢ and -Ti are, respectively, the pdf and the cdf of the standard normal distribution; thus, the limiting density fo,) is given by

fe(w) =

[1+q{220(q)

7r(1+w2)

1}, exp (mx+my)

which is the same density found by Marsaglia (1965, page 196, equation (5)) for the ratio of independent normal random variables with means (mx, my) and unit variances. The density f (w) = f (w; mx, my) may be confined for positive values mx > 0, my > 0 since (3.11) shows that

f (w; -mx, my) = f (-w; mx, my) , f (w; -mx, -my) = f (w, mx, my) and

f (w;'mx, -my) = f (-w; mx, my) . Figure 3.1 shows how variations in (mx, my, v) affect the shape of the density. It may be seen by reference to Marsaglia (1965) that the shapes are similar to that of the ratio of normal random variables, even for small values of v.

The percentage points of W defined by Pr(W < w) = p are tabulated in Press (1969) for cumulative probabilities p = 0.01, 0.05, 0.10, 0.90, 0.95, 0.99; v = 1, 2, 5, 10, 30; and for some 16 selected values of (mx, my). Here we provide the tables of w for (mx, my) = (0, 0), (mx, my) = (1, 3), (mx, my) = (3,1), and (mx, my) = (3, 3).

58

Linear Combinations, Products, and Ratios W,m*0 0

0

0

0 0 0

'

0

0

0 0

' I

a

-2

0

I

I

2

4

0

0

0 4

-2

0

T ----T -2

0

2

J

4

-2

I

-2

0

1

0

2

4

2

4

I

I

2

4

M-Jm1,m2:80

m2 3,my IA10

J

4

4a a

mr:I

4

2

I

I

2

4

J

-2

0

m>-Ar L1

a 0

1

11

i6 0 0

J

-2

0

2

4

T---T---l a

-2

0

Fig. 3.1. Densities of the t-ratio distribution (3.11) for (my, my, v) = (0,0, 1), (0, 0, 30), (1, 3, 1), (1, 3, 30), (3,1,1), (3, 1, 30), (3, 3, 1), and (3, 3, 30)

3.3 Ratios

59

Percentage points w for (m, my) = (0, 0) v

p = 0.01

p = 0.05

p = 0.1

p = 0.9

p = 0.95

p = 0.99

1

2

-31.820 -31.820

-6.314 -6.314

-3.078 -3.078

3.078 3.078

6.314 6.314

31.820 31.820

5

-31.820

-6.314

-3.078

3.078

6.314

31.820

-31.820 -31.820

-6.314 -6.314

-3.078 -3.078

3.078 3.078

6.314 6.314

31.820 31.820

10

30

Percentage points w for (my, my) = (1, 3) v

p = 0.01

p = 0.05

p = 0.1

p = 0.9

p = 0.95

p = 0.99

1

-10.254 -5.791 -1.331 -1.041 -0.681

-1.794 -0.902 -0.382 -0.312 -0.256

-0.721 -0.357 -0.166 -0.135 -0.109

1.321

1.120

2.394 1.795

10.853 6.842

1.018

1.486

0.951

1.288

0.922

1.211

6.464 2.892 2.350

2

5 10

30

Percentage points w for (m,,, my) = (3,1) v 1

2 5

10

30

p=0.01 p=0.05 p=0.1 p=0.9 p=0.95 p=0.99 -51.268 -58.076 -64.675 -67.570 -69.742

-8.970 -10.187 -11.427 -11.984 -12.406

-3.604 -4.034 -4.524 -4.756 -4.934

6.604 7.337

8.007 8.293 8.505

11.970 13.422 14.774 15.355 15.788

54.267 61.290 67.984 70.894 73.074

Percentage points w for (m, my) = (3, 3) v

p = 0.01

p = 0.05

p = 0.1

p = 0.9

p = 0.95

p = 0.99

1

-12.970 -8.172 -2.668 -0.118 0.130

-1.852 -0.474 0.229 0.334 0.384

-0.442 0.183 0.425 0.477 0.505

2.242 2.141 2.038 1.986 1.943

3.652 3.205 2.802 2.627 2.497

14.770 11.028 7.498 6.019 4.997

2 5

10

30

Linear Combinations, Products, and Ratios

60

Since it is evident from (3.11) that f (w) -* 0 rapidly as m,, and my become large, values of my, my greater than 3 are not considered. The t-ratio distribution has one or two modes, depending upon the values of the parameters. The location of these modes are solutions of the equation v + 1 I + A(w)

T,+1 C q*

/

14

= B(w),

v+3I

(3.12)

\ q*

where

_

it (W)

1

B(w) = 2 +

(v+1)q2 qw+mx l+w2 (q*)2

(v + 2)F (v/2)

A(w)

-+W2

3qw + rn

1

{

q

}

2F ((v + 1)/2) m1 + w2 + 3qw

2

+ r ((v + 2)/2)

fl' ((v + 1)/2) x

1 + w2

{ 1 + (q/q*)2 }

(v+2)/2

q* {m 1 + w2 + 3qw}

{w 1+w2 -q'mx

and q and q* are as defined above. Note that since T,, may be expressed in closed form in terms of elementary functions, (3.12) yields the modes in terms of elementary functions only. From (3.11),

lim q

-mX

and

lim q*

w-aoo

=

m2 + v.

Thus,

lim w2f(W) = Constant.

w-foo

Hence the distribution of W can have no finite moments of order above zero.

Kappenman (1971) extended Press's ratio distribution for the multivariate case by considering the joint pdf of WT = (X2/X1, X3/X1, ..., XP/X1), where X = (X1i ... , XP)T is a p-variate t random vector with degrees of freedom v, mean vector µ, and correlation matrix R. The

3.3 Ratios

61

expressions for the joint pdf turned out to depend on whether p is odd or even. Introduce the following notation

VT =

(1, WT),

M = VTR-1V,

K = -2VTR-1µ, L = v + µT R-1 µ, _

a

L

K2

M

4M2'

and

K

b

2M Then the expressions for f (w) are a 1111

f (W)

_

-1/2

yv12l' ((v + p)/2)

(p-11)`/2

p-1

Irp/2 (Ma)(v+p)l2 r (v12) O G-1-2k ) a x

J _00

u2k {au2 + 11

2k

by -1-2k

-(,+p)/2

du

ifpisodd; f (w)

2abp-1 JR1-1/2 vvl2r ((v + p)/2) .7rp/2 (Ma)(v+p)l2 F (v/2) P/2

1: k=0

(:) p

2k-i

\b1

oo X

J

b/a

u2k-1 {a2u2 +

(p-2)/2

+

p-1

> (p2ki) \

a}-(v+p)l2

du

a) 2k

k=0

x

u2k {a2u2 + a} -(v+p)12 du

Linear Combinations, Products, and Ratios

62

if p is even and b < 0; and f (W)

-

IR1-1/2 2abp-1 vv12r ((v + p)/2) 7rp/2M(v+p)/2P (v/2)

p/2

p_1

a 2k-1

F (P-) \b1 k=o X

r

Jb/a

u2k-1 {a 2u2 +a

(E-2)/2

+

k=O

p-1

I

du

(a)2k

(2k1) \bl 6/a

X

I - (v+p) /2

u2k {a2u2 +a) -(v+p)12 du

if p is even and b > 0. The integrals in these expressions can easily be rewritten in terms of the gamma and incomplete beta functions; see Section 3 of Kappenman (1971) for details.

4 Bivariate Generalizations and Related Distributions

In this chapter, we shall survey a number of specific bivariate distributions that contain Student's t components.

4.1 Owen's Noncentral Bivariate t Distribution Let Y1 and Y2 have the bivariate normal distribution with zero means, unit variances, and correlation 1. Let vS2 have the chi-squared distribution with degrees of freedom v and be independent of the X's. Then X 1 = (Y1 + 61) /S and X2 = (Y2 + 62) /S have the noncentral univariate t distributions with degrees of freedom v and noncentrality parameters 61 and 62, respectively. Owen (1965) studied the joint distribution of (X1, X2), a noncentral bivariate distribution. The marginal cdf of Xj, j = 1, 2 may be written as Pr (X j < _ y)

_ r(v/2)2(U-2)/2

\ fx1(x)(_8dx, Y_x

/

(4.1)

where 0 and 4 are, respectively, the pdf and the cdf of the standard normal distribution. Integrating by parts repeatedly, one obtains for odd values of v

Pr (Xj < y)

=

-D (-6jVB_) + 2T (6j IB, A)

+2 [M1+M3+ and for even values of v

Pr(Xj
64

Bivariate Generalizations and Related Distributions

Here,

A=

Y -

B = T(h a) = '

V

v+y2'

1 faexpf-(l 1+x2)h2/2} dx + x2

(4.2)

27r

(a function discussed and tabulated in Section 46.4 of Kotz et al., 2000), and the M's are defined recursively by

M-1 =

Mo = Av

0,

' (8,,/B) -D (6jAV-B)

,

+ AV Ml = B {8?AMo 27r l Vr-(6i) M2

B {8jAM1 +Mo},

M3 =

2B

16jAM2 + Mil,

and

Mk =

(k kl)B {ak5jAMk_1 + Mk_2},

k > 4,

where ak = 1/((k - 2)ak_1), k > 2, and a2 = 1. Two special cases of (4.1) are

Pr (Xj < 0)

=

(-8; )

and

Pr(Xj <1) =

1,

v=1.

Also, if 8; = 0, then (4.1) is just the cdf of the Student's t distribution. Owen (1965) expressed the joint cdf of (X1, X2) in terms of functions of the form R

Q (y, 8; 0, R) -

-

r(v/2)2(v-2)/2 Jo Jo

o0

()

xi-1 x

(

v

6 \\ dx

4.1 Owen's Noncentral Bivariate t Distribution

65

and

Q (y, 6; R, c) = r(v/2)2(--2)/2 f J0

R

/

oo

f

\

I T - S I dx.

0

\\l

v

/

For example, if one assumes, without loss of generality, that yl > y2, then the following relations hold

Pr(X1
Pr (Xi < y1, X2 < y2)

= Pr (X2 <- Y2),

for 61 > 62 and 61 < 62, respectively. The formulas for Q(y, 6; 0, R) and Q(y, 6; R, oo) can be obtained by integration by parts. Since Q(y, 6; 0, R)+ Q(y, 6; R, oo) = Pr(Xj < y), it is sufficient to know the formula for one of the Q terms. Owen obtained the following formulas for Q(y, 6; 0, R) for odd and even values of v, respectively

Q(y, 6; 0, R) = -(R) - 2T (R, (AR - 6)/R) -2T (8,/B, (6AB - R)/B8) + 2T (8,', A)

+I{6<0}-1+2{Mi +H1+M3 +H3

and

Q(y 8; 0, R)

= 4, (-8)+

27r {Mo +Ho +M2 +H2

Here, T(h,a) is as defined in (4.2) and the M's are defined recursively by

Mo = A/ (8,') {b (8A/) MI*

=

((6AB

- R)/\) }

B[6AMo +A0(8,1B) {0

-0 ((8AB - R)/./B)

M2 =

B

{BAMi + Mo } - L1

,

Bivariate Generalizations and Related Distributions

66 and Mk*

=

(k - 1)B k

{ak8AMk* _1 +

Mk-21 - Lk-1,

k > 3,

and the H's are defined recursively by

Ho = -c(R)4 (AR - S) and

Hk = ak+2RHk-1,

k > 1,

where

Lk-1

= ak+2RLk-2,

k>3

with the initial value L1 = (ABR/2)0(R)O(AR - 6) and

k>3

ak (k - 2)ak-1'

with the initial values a1 = a2 = 1.

4.2 Siddiqui's Noncentral Bivariate t Distribution Siddiqui (1967) considered the joint distribution of Student's t variates when (Y1i, Y2i), i = 1,.. . , N is a random sample from the bivariate normal distribution with zero means, unit variances, and correlation coefficient p. Let 1 N Y1

1'2

N

=N

Y1i,

i-1

N

Y2i, i=1

Si

=

1

N

N(Yii-Yi)2, i=1

S22

-

1

N

N

1: (Y2i - Y2)2 i=1

4.2 Siddiqui's Noncentral Bivariate t Distribution

67

and 1

R=

N

E(Y1i-YO (Y2i-Y2)

NS1S2 i=1

The interest is in the joint distribution of the Student's t variates

(X1, X2) =

Yi

Y2

( N-IS1' N-1S2

It is well known that the joint pdf of (Y1, Y2, S1, S2, R) is f (91, y2, S1, S2, r)

=

2 (N-4)/2

N N(802) N-2

(1 - r)

exp

27r2I'(N-2)(1-p2)N/2

+y2 + s2 - 2p (xy + rsi s2)

N 2(1-p2)

x2 + s21

}

for -oo < y1 < oo, -oo < 92 < 00, 0 < 81 < oo, 0 < s2 < oo, and -1 < r < 1 (see, for example, Kendall and Stuart, 1958). After suitable transformation, Siddiqui obtained the joint pdf of (X1, X2, R) in the form I' (v + 2) (1 - p2)

f (x1, x2, r)

=

(v+1)/2 (I

- r2) (v-2)/2

(27r)3//2/r (v + 3/2) (1 - b - cr) v+1l2

vl

X S 11+ x 2F1

I

(i+)} -(v+

1)/2

2

1 3 1+b+crl 2;v+2' 2 2,

C1

,

where v = N - 1, 2F1 is the Gauss hypergeometric function, b

_

Px1x2 V

1+

-+4 ,

and

c=

P

It is easily seen that the limit of this joint pdf as v -4 oo is trivariate normal with independent components (X1, X2) and R. Integrating out

Bivariate Generalizations and Related Distributions

68

r in (4.4), Siddiqui showed further that the asymptotic joint pdf of (Xi,X2) as v -i oo becomes r(v + 2) (1 - p2)("+1)/2

f (xl, x2)

-c2}1-'/2

{(1 -b)2

27r J F (v + 3/2) (1 + X2 1J

1

Sl

3

1-Jb2+c2

)

X 2F1 (21 2'v+2' 2(1-b)

The exact joint pdf of (X1, X2) was also given for the two special cases v = 1 and v = 3. For v = 1, the joint pdf reduces to a bivariate Cauchy distribution pl) { 1 + (2 B) cot B} , (4.5) f (xl, x2) -- 2 -(1+

-

x2) (1c+e 2)

where 2p (1 - p2) (1 + y1y2)

1+yi 1+y2

lx

In fact, if p = 0 in (4.5), then one arrives at a product of two independent Cauchy densities. For v = 3 the joint pdf is f (x1, x2)

2

-

p2) 351-2

(i+) (i+a) 2

2

2

j3,

where 13

r(9/2)r2(k + 1/2) = y., r(k+9/2)r2(1/2)k! J(1

1

1

2)

-05-21

- b - c)'-5/2 - (1 - b +

(k)

c)1-5/2 }

.

4.3 Patil and Liao's Noncentral Bivariate t Distribution Patil and Liao (1970) provided an extension of Siddiqui's work when (Yii, Y2i), i = 1, ... , N is a random sample from the bivariate normal distribution with zero means, common variance a2, and correlation coefficient p. Instead of considering the joint distribution of (4.3), they considered the joint distribution of

(X1, X2) _

SYI

,

SY2

,

(4.6)

4.4 Krishnan's Noncentral Bivariate t Distribution

69

where S is the pooled sample standard deviation defined by N

N

2

2

' - 1'1) + i=1 (Y 2i - Y2) .i.1 lYiz 2N - 1

S =

Exact expressions were given for the joint pdf and cdf of (X1, X2) and for the corresponding marginal distributions. For instance, if N is odd and equal to 2q + 3, then the joint pdf of (X1, X2) can be represented in the form f (X1, X2)

1

= 22q+sir(q + 1) (1

p2)q+3/2 r2(q + 1)

-

q

X

(q) k=O

(-1)q-k (.I_P

k

{ 2(1 + P) 1

1

p

2q-k 1=0

)

p

+ 2)

-1

2q-k+l

2

(1 -p2)

+

r(2q-k+1) y}_(k+2)

+

- 2Py1Y2

8(q + 1) (1 - p2)

F(k + l + 2) l!

1

2(1-p) -(k+1+2)

+ Y1

- 2Pyly2 + y2

8(q + 1)(1 - p2)

while if N is even, then the joint pdf of (X1i X2) becomes (1 + p)(1-N)/2 J (x1, x2)

_

,7r2N+l(N - 1)(1 - p) (N-1)12 X

k

((1 -N)12 ) k

1 - p2

}2 (N + k - 1) {

(

1

+ P) 2

- N +k

X

yi - 2Pyly2 + y2

1

{ 2(1 + p) + 4(q - 1)(1 - p2) }

4.4 Krishnan's Noncentral Bivariate t Distribution Krishnan (1972) provided another extension of Siddiqui's work when (Y1i, Y2i), i = 1, ... , N is a random sample from the bivariate normal distribution with means (-y, 8), unit variances, and correlation coefficient p. She considered the joint distribution of

=

(

N-1Y1 Sl

N-1Y2 S2

)'

(4.7))

Bivariate Generalizations and Related Distributions

70

where Sl and S2 are correlated chi-squared random variables independent from Yl and Y2, respectively. Series representations were derived for the joint pdf and the cdf of (X1, X2). One of the representations given for the joint pdf is

y

00

f (XI, X2)

= l

j=0

C1+N 11/

(1+N

21)

00

x E d,,,, m=0

where

A =

N 2 (1 - p2)'

A3NN24-N 3-N exp {-A (ry2 + j2 - 2p-y6) } 7rP ((N - 1)/2)

B

(2pAxlx2)2-k

yk Cj

(N -

1)j-k(j - k)!'

(pA) (N-3)/2+k / {(k/2)!P (N + k - 1)/2)),

Vk

0,

dm

,

if k is even, if k is odd,

(-2c)2(-2d)",-ZI,(N+i+j11,(N+j+m-al

= i=0

i!(m - i)!

2

C =

A (pa - 'Y) xl

d=

(p'Y -

J

2

J

N-1+xi

and S) X2

N-1+x2

In the central case -y = 8 = 0, (4.8) reduces to the form derived by Patil and Liao (1970).

4.5 Krishnan's Doubly Noncentral Bivariate t Distribution

71

4.5 Krishnan's Doubly Noncentral Bivariate t Distribution If Y is a normal random variable with mean S and unit variance, and S2 is an independent noncentral chi-squared random variable with degrees of freedom v and noncentrality parameter A, then

X =

Yv/-V-

S

is said to have the doubly noncentral univariate t distribution with degrees of freedom v and noncentrality parameters S and A. The properties of this distribution have been studied by several authors; see Robbins (1948), Patnaik (1955), Krishnan (1959), Krishnan (1967a), Bulgren and Amos (1968), and Krishnan (1968) - see also Chapter 31 in Johnson et al. (1995) for a summary. The pdf, the expectation, and the variance of X are given by 00

ao

E k=01=0

Ak exp {- (A + 52) /2} Fv2kk!17 (1/2 + 1) Be (1/2 + 1/2, v/2 + k)

2v) (1 + E(X) =

v-1

v S

2r (

2

1

)

1F1

-(v+1+2k+1)/2

x2

Sx

v

V

)

vl r \21'

a

(2' 2' 2)

v > 1,

and

Var(X) =

(1

V

+ S2)

v-2

1F1

(1, 2;

- 2) - {E(X)l2

,

v > 2,

where 1F1 denotes the confluent hypergeometric function. A bivariate analog of (4.9) was defined by Krishnan (1970) as follows. Let (Y1, Y2) follow a bivariate normal distribution with zero means, unit variances, and correlation coefficient p. Let (S1, S2) follow independently a noncentral bivariate chi-squared distribution with degrees of freedom v, noncentrality parameter A, and correlation coefficient p (Krishnan, 1967b). Then the random vector (X1,X2)

Y2-1,5

Sl ' (Yip

Sz

)

(4.10)

is said to have the doubly noncentral bivariate t distribution with degrees of freedom v and noncentrality parameter A. Krishnan (1970) derived

72

Bivariate Generalizations and Related Distributions

the corresponding joint pdf of (X1, X2) and provided an application involving the sample means and variances from two correlated nonhomogeneous normal populations. The special case of (4.10) for S1 = S2 = S was considered by Patil and Kovner (1969), who provided expressions

for the joint cdf of (X1i X2) and showed that when the means of Yj are zero the probabilities of (X1, X2) in rectangular regions are monotone functions of p. In the special case Si = S2 = S and A = 0, the distribution of (X1, X2) reduces to that of the central bivariate t.

4.6 Bulgren et al.'s Bivariate t Distribution Suppose Y1,. .. , Y,,,,, Y,,,,+1i ..., Ym+n denote iid normal random variables

with common mean p and common variance a2. Bulgren et al. (1974) considered the joint distribution of (X1, X2) defined by (X1,X2)

m+nY2

mYl

=

S2

n-1)S; m+n-2 )S2

(

where 1

Y1

m

=

i=m+1

mY1 + n,,*

Y2

S21

-

m+n

..,

E(Yi-Y1)2 1

i=1

and 1

S2

n-1

m+n

(Yi-Y*)2 i=m+1

The distribution of (X1, X2) is bivariate t with a different noncentrality

parameter for each variable. Note also that X1 and X2 have, respectively, m - 1 and m + n - 2 degrees of freedom. Bulgren et al. (1974)

4.7 Siotani's Noncentral Bivariate t Distribution

provided series representations for the joint pdf of (Xl, X2). central case µ = 0, 2 (m+n)/2 A

f (x1, x2)

xE1)3r(m+n+

(1+ 7

j!

\k)B

2

'

l2

\n(m-1))

2j-k

-_x2

xxi

n+m

/

2

)-

x2

m+n-2

)

\m+k n-1

2 (2j xk=o

In the

-(m+n)/2

2 2

Cl+m+n-2

j=0

73

(4.11)

n(m+n-2)

where

A=

2(m+n)/27f

n(m-1)(++ n-2)1, I m2 1)I'I n-2

1

In the noncentral case p $ 0, the joint pdf is `even more complicated. Letting n = am, a > 0, and taking m -+ oo in (4.11), one observes that the limiting distribution in the central case is the bivariate normal distribution with zero means, unit variances, and correlation 1/(1 + a).

4.7 Siotani's Noncentral Bivariate t Distribution Siotani (1976) considered the most general form of (4.6) introduced by Patil and Liao (1970). Let Y be a p-variate normal random vector

with mean vector p, unit variances, and correlation matrix R. Let S = (V2 +V2)/(2v), where (VI,V2) has the central bivariate chisquared distribution with degrees of freedom v and correlation coefficient T. Siotani derived the distribution of X = Y/S for general p and R. The

derivation required the joint pdf of (V1,V2) that was given by Siotani (1959) in the form cc

f (V1, V2)

=

v+2k2)(v+2k2/v+2k

1: Ck(T) k=O

1

(V1

(1-T )

2

x exp { l

r2 ((v + 2k)/2)

V1 + v2

2(1-T2)

,

where C k (T)

=

F (

1T

' X2)/2) (1 -

T2)v/2 r2 k.

(4.12)

Bivariate Generalizations and Related Distributions

74

From this, one can easily obtain the pdf of V1+V2 2v (1 - T2)

S

W=

T2

as

f (w)

Ck(T)f2v+4k(W), E k=0

_

(4.13)

00

where

=

f2v+4k(w)

2(2v) v+4k

2k)w2v+4k-1

2v+4kr(v +

exp {-vw2} .

(4.14)

Since cl(T) + + co. (T) = 1, (4.13) is a mixture of (4.14) with the weights given by (4.12). Thus the joint pdf of X is also obtained in the same form 00

Ck(T)f2v+4k (x)

,

k=0 where Ck(T) are given by (4.12) and

f+4k (x) = exp

{_,2TR,} r (v + 2k + p/2) (1 2

(2vx)p/2r (v + 2k) IRI112

x{1+

1 - T2 xTR-1x} - (v+2k+p/2) 2v

0 r (v + 2k + (p + l)/2) l!r (v + 2k + p/2) 1=0

L,

X

2 (1 - T2)xTR-1µ 2v + (1 - T2) xTR-1x

X

When p = 2, µ = 0, and p = T (p is the correlation coefficient between Y1 and Y2) this coincides with the pdfs derived by Patil and Liao (1970).

4.8 Tiku and Kambo's Bivariate t Distribution Suppose (X1, X2) has the bivariate normal distribution with means (Al, µ2), variances (o, , a2), and correlation coefficient p. Its joint pdf can be factorized as f (x1, x2)

= f (x1

I x2) f (x2)

4.8 Tiku and Karnbo's Bivariate t Distribution

75

where

f(xiIx2) =

1

Ql

1 - p2

expl-

1

z z 2Q1 (1 - p)

al

i

x1-µl

z

(4.15)

20'2 (x2 - µz)2

(4.16)

-p-92(x2 - µ2) and

f (x2)

1

a U2

exp S l

z

Numerous nonnormal distributions can be generated by replacing either f (xl I x2) and/or P X2) by nonnormal distributions. Tiku and Kambo (1992) studied the family of symmetric bivariate distributions obtained by replacing (4.16) by the Student's t density f (x2)

a

(x2 -

µ2)2

1+

V

(4.17) kor2

where k = 2v - 3 and v > 2. This is motivated by the fact that in many applications it is reasonable to assume that the difference Y1 - µl P(91/92)(Y2 - µ2) is normally distributed and the regression of 11 on Y2 is linear. Moreover, in numerous applications Y2 represents time-tofailure with a distribution (Tiku and Gill, 1989; Gill et al., 1990), which might be symmetric but is not normally distributed. Besides, most types of time-to-failure data are such that a transformation cannot be performed to impose normality on the underlying distribution (see, for example, Mann, 1982, page 262). On replacing (4.16) by (4.17), the joint pdf of (Xl, X2) becomes }_V

f (XI, X2)

=

1

Qloz

{i+ (X2 - µ2)2 ka2

k(1 - p2)

x exp -

1

2Q1

1

x1 - µl -P - (x2 -µ'2)2

(1 - p2)

z

1J

(4.18)

Limiting v -+ oo, (4.18) reduces to the bivariate normal pdf. Writing p2j = E((Y1 - µ1)2(Y2 - µ2)j) for the cross product moment of order i + j, one observes that all odd-order moments are zero and that the first few even-order moments are 2

76

Bivariate Generalizations and Related Distributions P11

= P6192,

/

= 3al I

µ4o

=

P31

P22

/43

=

3pal

2P4

1 + 2v

5

I

22 (i+ 2v-5

a2

P

)

I

6 P2

cic2 (1+2 p2 +

= 3pala23

(1 +

2v - 5 2

2v '5

'

and µ04

=

3(2v - 3) 2v - 5

4

a2

In fact, the moment generating function (mgf) of (Y1,Y2) is given by E [exp (01X1 + 02X2)]

= exp I Al - pal µ2 101 +

(1

81 12

2

OF2

xM2 02 +Cpl a µ201J

_2

,

92

denotes the moment generating function of X2. This moment generating function does not exist unless, of course, v = oo. However, the characteristic function does exist and is given by Sutradhar (1986). Estimation issues of the distribution (4.18) are discussed in Section 10.1. where

4.9 Conditionally Specified Bivariate t Distribution Let (X, Y) be a continuous random vector with joint pdf fx,y (x, y) over R2. Let fx(x), fy(y) and fxI y(x I y), fy, x(y I x) denote the associated marginal and conditional densities, respectively. Assume that X I Y and Y I X are Student's t-distributed with the pdfs

fxIr(x I y) =

1'((v + 1)/2) V"7-1'(v/2)

V/0(y) {1 +

or(y)x2}-(-+1)/2 (4.19)

4.9 Conditionally Specified Bivariate t Distribution

77

and

r(r(v/2)

fyix(y I x) =

+r(x)y2}-(v+1)/2

T(x)

(4.20)

where x E 2, y E Q, v > 0, and a(y), r(x) are some positive functions. Writing the joint pdf of (X, Y) as a product of marginal and conditional densities in both possible manners, one obtains

fr(y)

a(y) {1 + Q(y)x2}-(u+1)l2

= fx(x) r(x) {1 +T(x)y2

(4.21)

where x E R and y E R. Set 2/(v+1)

2/(v+1)

g(y) _{fr(y)

h(x) _ {fx(x) T(x)}

ma(y)}

(4.22)

so that, after rearranging, (4.21) becomes g(y)

+ y2g(y)T(x) - h(x) - x2h(x)o, (y)

=

0,

(4.23)

which must be solved for a, T, g, and h. Kottas et al. (1999) recognized that (4.23) is a special case of the functional equation n E fk(x)9k(y)

=

0,

k=1

whose most general solution is given in the classical book by Aczel (1966,

page 161). Thus, with h(x), x2h(x) and g(y), y2g(y) being the systems of mutually independent functions, the solution of (4.23) is found to be T(x)

_

_

_3 + _4x2 Al + A2x2'

a(y) =

A2 + A4y2

(4.24)

Al + A3y2

and (x)

h

1

1

=

Al + A2 X2'

9(y) =

(4.25)

Al + A4y2

for A3 E R, j = 1, 2, 3, 4. Finally, substituting (4.22), (4.24), and (4.25) into (4.21), the joint pdf is derived as

fx,y(x,y) = Nv(A1, A2, A3, A4) x {A1 + A2 X2 + A3 Y2 +

A4x2y2}-(v+1)12

, (4.26)

denotes the normalizing constant. Utilizing certain compatibility conditions given in Arnold and Press (1989), Kottas et al. found that (4.26) is a well-defined joint pdf if Al E R+U{0}

where x E 1, y E l1, and

78

Bivariate Generalizations and Related Distributions

and Aj E R+, j = 2,3,4. Moreover, if Al = 0, then one must have v E (0,1). The normalizing constant is given by the integral 1

Nv(A1,A2,A3,A4) 00 A4x2y2)-(v+l)/2

(A1 + A2x2 + A3 Y2 +

J7.

dxdy. (4.27)

00

In the case Al 54 0, making the transformation s = (A2/A1)x2, t = (A3/Ai)y2, letting 0 = A1A4/(A2A3), and using the integral representation of the Beta function, B(a, b)

=

J0

a> 0,

o" xa-1(1 + x)-a-bdx,

b> 0,

one obtains

_

1

Nv(A1, A2, A3, A4)

B (2'

dx

A(v-1) /2 2)12A3

(1 + x)v/2

0

x(1 +

Ox)

Letting w = x/(1 + x) and manipulating, Kottas et al. obtained v

Nv(A1, A2, A3, A4)

=

2

B (21, 2L) 2)

1

(1

2 31 /1

12' 2'

(4.28)

2

where 1

I (a, b, c; x)

=

L

for c > b > 0. In the case Al = 0, similar arguments show that

N(0

,

A 2, A 3, A 4 )

(A2A3)v/2A41-v)/2 _ B(1 vB(1=v v,

2'2

2

'21

where 0 < v < 1. The integral (4.29) converges for z < 1. For I z I> 1, Kottas et al. provided an alternative representation of (4.28) in terms of the Gauss hypergeometric function (see, for example, Magnus et al., 1966, page 54). It is also possible to represent (4.28) in terms of elliptical

integrals of the first and second kind (Carlson, 1977, Chapter 9). For example, if v = 1, then (4.28) can be easily rearranged to yield Nv(A1,A2,A3,A4)

=

V1 A4

27rRF(0,1/0,1)'

4.9 Conditionally Specified Bivariate t Distribution

79

where RF is the elliptical integral of the first kind defined by RF(a, b, c)

=

2

f00

{(x + a)(x + b)(x + c)}-1/2 dx

J0

with a, b, c nonnegative, and at most one of them equal to zero.

If 0 < v < 1, then (4.26) does not possess finite moments; thus, from here on we shall consider the case v > 1. If v > max(m, n), for non-negative integers m, n, then Kottas et al. showed that

E (X '"Y"`

=

r(

)2

)r(v m2

)I(

2

1 1 , V+1; [11"[12 r (L) V) I (1, 2 2 2

;1-0) -(4-30)

- 0)

provided that both m and n are even or zero. The expectation is zero if at least one of m or n is odd. This suggests that the distribution may be an appropriate model for uncorrelated but nonindependent data. From relations (4.21), (4.24), and (4.26) it is immediate that the marginal densities are

fx(x) =

11

{I

(1,

11

v 22,

2

- 0)1

+µix2)v/2}-

1 +Oµ1x2(1

and fY(y)

=

1 µ2{I 22 v21;0) 1+Oµ2y2(1+µ2y2)vl2}

where x E l and y E R. Here, µj _ Aj+1/A1, j = 1,2 are the intensity

parameters while ¢ and v are the dependence and scale parameters, respectively. It is easily noted that X and Y are independent if and only if 0 = 1. The graph of the joint pdf is symmetric and bell-shaped and takes the standard form when µ1 = µ2 = 1. From relations (4.19)-(4.20) and (4.24)-(4.25) it is immediate that X I Y has the Student's t distribution with degrees of freedom v and scale parameter (1/v1)(1 + µ2y2)/(1 + 0µ2y2), and that Y I X is also Student's t with degrees of freedom v and scale parameter (1/v2)(1 + µ1i2)/(1 + Oµ1x2), where µi = .j+1/)1, j = 1, 2. Consequently, the conditional moments are

E(Xm I Y=y) =

r (m+l) r (v2

2

V'571' (2)

1 +µ2y2

m/2

µ1 (l + Oµ2y2) /

and

E(Y" I X = x) =

r(I)r(v=m) 2

V/7rr(2)

l+[11x2 2 µ2(1 +Oµ1x)l

m/2

80

Bivariate Generalizations and Related Distributions

provided that m is an even number less than v. If m is odd, then the corresponding conditional moments are zero. In the special case v = 1, (4.26) reduces to the centered Cauchy conditionals model of Anderson and Arnold (1991). The limiting case 0 -4 0 gives the bivariate Pearson type VII distribution (Johnson, 1987, page 117) with location parameters equal to zero and uncorrelated components. If 2v is a positive integer, then this limit distribution reduces to a special case of the general bivariate t distribution (see, for exam-

ple, Johnson and Kotz, 1972, page 134, relation 1) with uncorrelated components and 2v degrees of freedom. For p, = µ2 and v = 2, the limit distribution reduces to the bivariate Cauchy distribution (Mardia, 1970a, page 86) while for µl = 42 and v = v + 1 it gives the bivariate t distribution (Johnson and Kotz, 1972, page 134, relation 2) with v degrees of freedom. In the latter case, the standard bivariate normal distribution with independent components arises as a further limiting case when v -+ oo. Other special cases of (4.26) are the centered normal conditionals model studied by Sarabia (1995) and the Beta conditionals model of the second kind (Castillo and Sarabia, 1990).

4.10 Jones' Bivariate t Distribution Let Z1, Z2, W be mutually independent random variables with Zi having the standard normal distribution and W having the chi-squared dis-

tribution with degrees of freedom n1. Then the standard bivariate t distribution with degrees of freedom n1 is the joint distribution of n1Z2

Vn1Z1 W_

I

VV_

)

(4.31) *

One disadvantage of this model is that the two univariate marginals (which are Student's t) have the same degrees of freedom parameter and hence the same amount of tailweight. Jones (2002b) provided an alternative distribution with Student's t marginals, each with its own arbitrary degrees of freedom parameter. Precisely, if W1, W2 are independent chi-squared random variables (also independent of Z1, Z2) with degrees of freedom v1 and v2 - v1, respectively, then Jones (2002b) considered the joint distribution of (X1, X2)

=

v1Z1

v2

Z2

(4.32)

VIWI ' V' Wl +W2 Note that the ith marginal of this distribution is Student's t with degrees of freedom vi. It is easy to see that the correlation between X1 and X2

4.10 Jones' Bivariate t Distribution

81

is zero, a property also shared by (4.31). If rl < vl and rl + r2 < v2, then the product moment is given by

E(Xi'X22) _

vii/2v22/2r(l+ri)r(l+rz)r(vi-rl\r/v2-rz) 2 2

2

-2

J

7rr(2)r( 2

1

if rl and r2 are even and is zero otherwise. The joint pdf of Xl and X2 is

f (xl, x2)

x2 \ -(1+v2/2)

x2

CI1+vi+1121 1 + 112 . X 2 v

v2

v2 -

2F1 1+ 2

2

2

Cl +

`

x12 V1

y1

2

+

2 112

(4.33)

/

where C

r (1+V1 ) r (

1

-

7f

V217 2

2

and 2F1 denotes the Gauss hypergeometric function. The conditional pdf of X2 given Xl = xl is x2) -(1+L 2/2)

f (x2 I x1)

C

(ul + 21 112

/

X 2F1 11+

ul-1

112 V2-v1 1+112

\\\

2'

2

'ul+x2/ 2V2)

2

(4.34)

where ul = 1 + x 21v, and

C =

u(l+vl)/2r (1 + 7fv2

(l+v \z 2

)

If v2 + 1 > r, then the conditional rth moment is given by E

(Xr

2

I X, = x l) = 1

yr/2u(r-V2+vl)/2 2 1

r(

r ( 1+v2-r ) \ "r( ) 2

)

2

1+112 2

X 2 F1

C

v2-r+1 112-vl 1+112 U1 2

2

2

'

1)

U1

(4.35)

if r is even and is zero otherwise. Setting 112 = vl = v in (4.34)(4.35), one obtains the corresponding forms for the standard bivariate t distribution (see Section 1.11). Note that the conditional variance

Bivariate Generalizations and Related Distributions

82

Var(X2 I X1 = x1) increases with I x1 I. In a parallel fashion, with u2 = 1 + x2/v2, the conditional distribution of X1 given X2 = x2 is

f (xl

I X2)

= C

-(1+vi/2)

x2

u2 +

V11

U2 V2-v1 1+1/2

X 2F1 1+

2'

2

2

U2 1

u2 + X211V1

where

C=

u(1+v2)/2r

(1) r' (2) r (1 + 2 ) Vrvlr(z)r(1 )

This time, the conditional rth moments exist provided v1 > r, unless v1 = v2, in which case one needs 1 + v1 > r. The odd conditional moments are again zero and the even conditional moments are given by the simpler form r E' (X1 I X2 = x2)

r/2u2r/2r (12r) r (v) r (2) r

=

v1

(v2

2+1)

r (2) r (1±v2) r () 2

2

The construction (4.32) can be easily extended to the multivariate case. Two straightforward extensions are

Let Z1,.. . , ZP, W1, ... , WP be mutually independent random variables with Zi having the standard normal distribution and Wi having the chi-squared distribution with degrees of freedom vi - vi-1. Then, (Xi, X2, ... , Xp)

-

Z1

v2 Z2

v1

W1

+W2'...,

' 11W1

vP Zp W1 +...+WP

(4.36)

has a multivariate distribution with univariate marginals that are Student's t distributed with degrees of freedom vi, i = 1,.. . , p. The bivariate marginals of (4.36) have the distribution of (4.32). With the notation as above,

(X 1,XZ,...,X'p)

-

v1 Z1 W1

v2 Z2

W1 + U2 '

... '

vP ZP

W1 + Up

(4.37)

has a multivariate distribution with the same univariate and bivariate marginals. Here, Ui are independent chi-squared random variables

(also independent of Z,, Wi) with degrees of freedom vi - v1, i = 1,...,p.

4.10 Jones' Bivariate t Distribution

(b)

(a)

-2

0

83

1

2

-2

0

1

2

X1

X1

Fig. 4.1. Jones' bivariate skew t pdf (4.33) for (a) vi = 2 and v2 = 3; and (b)

v1=2andv2=20

Further extensions of (4.32) arise by adding further independent chisquared random variables inside the square roots in the denominators of all variables in (4.36) or by adding a single further independent chisquared random variable inside the square root in the denominator of Xl in (4.37). Jones (2002a) provided another bivariate generalization of (4.31). This generalization has the skew t distribution (Jones, 2001a) as its marginals. If U denotes a standard beta random variable with parameters a and c, then a skew t variate is defined by

X-

a + c(2U - 1) 2 U(1 - U)

84

Bivariate Generalizations and Related Distributions

The corresponding pdf is

f(x;a,c)

-

21-a-c17(a+ c)

x

) )1+

a -+c I(a)I(c)

l a±II z

a + c + x2

}

}C+i/2

r

x{ 1+

(4.38)

a+c+x2

111

The standard Student's t is the particular case for v = 2a when a = c. If a > c, then (4.38) has positive skewness; also, f (x; b, a) = f (-x; a, b). Further details about (4.38) are given in Jones and Faddy (2002). The bivariate generalization proposed in Jones (2002a) is constructed in the same way as (4.38): Specifically, if (U,V) denotes a Dirichlet random vector with the joint pdf f(u,v)

=

r(a + b + c) ua-lvb-1(1

r(a)F(b)r(c)

- u -

v)c-1

(where u > 0, v > 0, and u + v < 1), then define (X1,X2)

-

(4.39)

1 - U ' 2 U1 12U

2

U where d = a + b + c. It can be verified that the joint pdf of (X1, X2) is

f (xl x2) =

dF(d + 1)

2d-1r(a)r(b)r(c)

a-1

xl

1+

d+x1 b-1

1

x

1

X2

V

d + x2 2

x2

d+x2

-

1

(d + x1)3/2 (d + x2)3/2

xl

d+xl

C

(4.40)

Because of a direct analogy with the Dirichlet distribution, only one of the two marginals of (4.40) can be a symmetric Student's t distribution, the other necessarily being skewed. This Student's t distribution will have degrees of freedom d, and any skew t marginal will have a total

parameter value of d, but divided up into unequal amounts. In this sense, marginals of (4.40) are most closely associated with Student's t distributions with degrees of freedom d. Note that if instead of (X1, X2) in (4.39) the transformation was made to (-X1, X2), then one would have obtained the equivalent distribution on x1 + x2 > 0. Also, (-X1, -X2) would have given the equivalent

distribution on x2 < x1 and (X1, -X2) as the same on x2 + x1 < 0.

4.10 Jones' Bivariate t Distribution

85

(b)

(a)

Nx

0

8

X1

X1

(C)

(d)

12

N

(0

N

N Nx

4

NX

N

X1

xt

Fig. 4.2. Jones' bivariate skew t pdf (4.40) for (a) a = 1, b = 1, and c = 1; (b)

a=3,b=4,andc=5;(c)a=5,b=1,andc=1;and(d)a=1,b=5, and c=1

The corresponding changes to (4.40) would simply have been to make corresponding changes to the signs of x1 and x2. The means and variances associated with (4.40) can be easily obtained from the results provided in Jones (2001a)

E(X

1

)

_

vfd-

(a

(b + c - 1/2) -b -c)r (a-1/2)r r(a)r(b + c)

(a

- b - c)

2

E(X2 ) =

Vd2

r (a + c - 1/2) r (b-1/2) r(a + c)r(b)

Var(XI) = d(a-b-c)2+d-2 _{E(X1)}2,

4 (a-1)(b+c-1)

86

Bivariate Generalizations and Related Distributions

and Var (X2)

=

d (a - b + c)2 + d - 2

4 (a + c - 1)(b - 1)

_ {E (X2)}2

.

The covariance between X1 and X2 appears not to be available in closed form.

5

Multivariate Generalizations and Related Distributions

This chapter contains a large number of modifications and extensions of the standard multivariate t distribution introduced in (1.1). Some of them are of somewhat complex nature. It thus requires a careful reading to see the forest behind the trees!

5.1 Kshirsagar's Noncentral Multivariate t Distribution one of the earliest results in the area of noncentral multivariate t distributions is that due to Kshirsagar (1961). Let Y be a p-variate random vector having the normal distribution with mean vector µ, common variance a2, and correlation matrix R. Let S2 be distributed independently of Y according to a chi-squared distribution with degrees of freedom v. Kshirsagar (1961) considered the distribution of X = Y/S and showed that it has the joint pdf .f (x)

= exp {

l

r ((v +p)/2) J (vx)pl2r(v/2) JR1 "2

2

l

(

x{1+1xTR-1x} ll

00

E where

v

(-+p)/2

JJJ

r ((v + p + k) /2) k!r ((v + p)/2)

v/2_xTR-1

v + xTR-lx

k ,

(5.1)

= µ/a. This noncentral distribution reduces to the form of

(1.1) when µ = 0. We noted earlier in Section 1.11 that, if X has the central multivariate t distribution with degrees of freedom v and correlation matrix R, and 87

88

Multivariate Generalizations and Related Distributions

if

X= and

( R11

R

R21

R12 1

R22 ) '

where X1 is p1 x 1 and R11 is pi x p1, then

Z1 = x1 and

Zz =

v+p1

1/2

11+ X1 R111X1 J

`

V

(x2 -R21R111X1)

v

are independently distributed, each according to a central multivariate t distribution. This result does not remain true to the noncentral distribution (5.1). Actually, Siotani (1976) showed that if

is the partition of l; corresponding to that ofCCX, 52.1 = b2 - R21R111S1 and

R22.1 = R22 - R21R1i R12'

Z2 is then the joint pdf of Z1 and(_TRi1'1 .f (Zl,Z2)

= Kexp

C

tt - 1'ttT 1Rz21SZ.1

xtv (zi; R11,pi) tv+p, (Z2; R22.1,p - p1)'

where the last two terms denote the pdfs of central multivariate t distributions with appropriate parameters and K is given by the formidable expression

K = r 00 IF ((v+p+k+l)/2) (2)k/2 ( 2 1/2 FF- k!l!F((v+p)/2) v v+p-p1 k=O c=o k

ZTR-% 11

I

5.1 Kshirsagar's Noncentral Multivariate t Distribution

(z R22

X

t

89

152.1)l

pl)}(k+l)/2

tt {1 + z2R22 1S2/(v +

Siotani (1976) also derived the corresponding noncentral distribution when X is partitioned into k sets of variates as in (1.20). Following the notations defined by (1.21), (1.22), (1.23), (1.24), and (1.25), the joint pdf of

= X1

Z1

and

=

Z1+1

v

+

ql

I 1 + 1 XTR1 1X(1)1

- 1/2

- R(II 1)TR-IX(l))

(x(1+1)

is given by the lengthy expression

f (Z1,...,Zk)

exp

(o? I

2 1=1

(v + 41-1) r ((v + ql-1)/2)

11=1

-(-+ql)/2

-1 ZIT

k

1+

X

r((v+gl)I2) IRII.(1-1)I-1/2

k

k

v+ql-1

l=1 00

00

11=o

Ik=0

r((v+p+l1+...+lk)/2)

X

T, ((v +p)/2) t 11...1 k )Im/2

X

k

m= 1 k

2 ( v + qm-1

z1 R 1

tt

m

X

M=I (1 + ZrRmm.(m-1)zm) where

_m - R(m) (m 1) TR-1 (m-1) (m-1) and 62

-

Sr(m-1)Rmm(m-1) m(m-1)

1m

1

Multivariate Generalizations and Related Distributions

90

5.2 Miller's Noncentral Multivariate t Distribution Let Y have the p-variate normal distribution with mean vector µ and correlation matrix R > 0. Let S be distributed independently of Y according to a v-variate normal distribution with mean vector A and correlation matrix mIp. Miller (1968) considered the joint distribution of

XT =(XI,X2,...,XP)= C1Si1,l52I,...,ISPI I, which he referred to as the generalized p-dimensional t random vector. Assuming I S 12 has the chi-squared distribution, Miller showed that the joint pdf of X is given by 21-(v+p)/2m-v/2F(v

f (x)

_

r(v/2)-P/2

+p) (!+XTR_1X) m

JRI1/2

T _1 x exp -1 µR µ }exp

xD_(v+p)

-(v+p)/2

m (xTR lµ)2 4 (mxTR-ix + 1) }

xTR-1 µ mxTR-ix + 1

(5.2)

'

where D_(v+p)(.) is the parabolic cylinder function (see, for example, Erdelyi et al., 1953). If µ = 0, then (5.2) reduces to

f (x)

=

+ p)/2) (m F(v/2)irP/2IRI1/2 1 + xTR m vl2r((v x exp

2m

iFi

v

-

(v+p)12

1x

+P V 2

IAIz

' 2' 2m (mxTR-1x + 1)

where 1F1 is the confluent hypergeometric function (see, for example, Erdelyi et al., 1953). If both µ = 0 and A = 0, then (5.2) reduces

to the usual central multivariate t distribution (1.1) with degrees of freedom v and correlation matrix R. To the best of our knowledge, this interesting distribution given by (5.2) has not been pursued further since its introduction some 35 years ago.

5.3 Stepwise Multivariate t Distribution Let Y be a p-variate normal random vector with mean vector µ, common variance a2, and correlation matrix R > 0. Let vS2 /a2 be a chi-squared

5.3 Stepwise Multivariate t Distribution

91

random variable with degrees of freedom v, distributed independently of Y. Then the joint distribution of Y1

X1 =

S

and Yk

Xk

72

r(1_V

-r(k_1)Rk11Y(k -1) S

J1+i_YT S2 k_1)Rk11Y(k-1), k = 2,...,p,

(5.3)

where rk denotes the multiple correlation coefficient between Yk and (Y1,...,Yk-1), Y(k)

R(k)

=

(Y1,Y2,...,Yk), 1

r12

"'

rkk

r21

1

...

r2k

rkl

rk2

_

"

1

and

r(k)

=

(rl,k+1, r2,k+1, ... i rk,k+1)

is known as the stepwise multivariate t distribution. This distribution has applications in linear multiple regression analysis; for instance, suppose that a random sample Y 1 ,. .. , Y,,,, corresponding to some nonrandom values (zii, z2i), i = 1, ... , n, is available. The null hypothesis to be tested is that the slopes of the two simple regression lines, Y on z1 and Y on z2, are both zero. Then, the X1 and X2 above could correspond to the usual t statistics for testing the two regression coefficients (Steffens, 1974). Steffens (1969a) studied the distribution of (5.3) for the special case R = Ir, the p x p identity matrix. In this case, since 1k = 0, r(k_1) = 0 and R(k_1) = I(k_1), (5.3) reduces to

Xk =_

/v+k_1Yk//1yTy

(k_1 )

92

Multivariate Generalizations and Related Distributions k-1

Vv+k-1S

2

Y S

+

Yk

v+k - lYk vS2 + Xl +

+ Xk_1

If µ = 0, Steffens (1969a) showed that X1, X2, ... , XP are independent Student's t random variables with degrees of freedom v, v+1.... , v+p-1, respectively. This result also holds for general R (Siotani, 1976, Corollary 3.1). In the noncentral case µ # 0, the XD's are still independent, but X1 has the noncentral distribution with degrees of freedom v and noncentrality parameter ul /a while the XD's (j = 2,3,.. . , p) have the doubly noncentral t distributions with degrees of freedom v + j - 1 and noncentrality parameter p, /u in the numerator and (µ1 /Q)2 + (p2/0,)2 + + (µi_1 /U)2 in the denominator. Steffens derived the joint pdf of the X3 .'s in the bivariate case as the double infinite series f (X1, X2)

=

exp{- (62 +62) /2} v(v + 1)F (v/2)

oo

00

1

(2

xk k!

V+1

l!

vI (v 62)k

(v + 1)1/(2k)

x2 ( I+l

(v+k+1+2)/2

+ x22

x2 x-

(y251)l

v1/(21)

k=O 1=0

v xI'+2k+21+1/ 1

-

-(v+k+1)/2

where 6j = µa /v, j = 1, 2. For general p and R, Siotani (1976) showed that if µ = 0, then X1, X2, ... , XP are still independent Student's t random variables with degrees of freedom v, v + 1, ... , v +p - 1, respectively.

In the noncentral case, the joint pdf of the Xj in (5.3) generalizes to f (xl,...,xp)

_

1

P

F ((v + k)/2)

P

2

2kTk 11 v+k-1F((v+k-1)/2) k=1 k=1

exp

P

x

x2

k 1+v+k-1

-(v+k)/2

1=1 CO

°O

x E ... k1=0 P

X

l=1

kp=0

F((v+k1 ++kp)/2) F ((v + p)/2)

2

v+l-1

k,/2

kl!...,p.

D

5.4 Siotani's Noncentral Multivariate t Distribution

1+x

93

v+l-1

where the rk's are the noncentrality parameters given by Tk

-

k

- r(k-1)R(kl 1)S(k-1) r 2k

and i;j = pj/o-.

5.4 Siotani's Noncentral Multivariate t Distribution In Section 4.5, we discussed a bivariate generalization of the doubly noncentral univariate t distribution given by (4.9). Siotani (1976) provided a multivariate generalization of (4.9) by observing that the pdf of S* = S// (where S is a noncentral chi-squared random variable with degrees of freedom v and noncentrality parameter A) can be expressed as the Poisson mixture CO

f (s*)

=

1: Pk(A)fv+2k (S*) k=0

where Pk(A)

_

exp(-A)Ak k!

is the kth probability of the Poisson distribution with parameter A and fv+2k (S*)

2v ("+2k)/2

=

(S*)v+2k-1

2(v+2k)/2f ((v + 2k)/2)

1

exp

ll C-2vS*2J

0

He defined X = Y/S* to have the doubly noncentral multivariate t distribution, where Y is a multivariate normal random vector with mean

vector µ, unit variances, and correlation matrix R. The joint pdf of X is easily obtained as a Poisson mixture of the noncentral pdf (5.1) with v + 2k in place of v in the arguments of gamma functions and in the power of 1 + xTR-lx/v, that is, P X)

=

0" k=0

Pk (A) fv+2k (x) ,

(5.4)

0

Multivariate Generalizations and Related Distributions

94

where

I ((v+2k+p)/2) fv+2k(x)

l

T

2

(v7r)rl2r ((v + 2k)/2) IRI1/2

l

r

(v+2k+p)/2

x S I+ ixTR-1x } v

11

)))

k

Vf2-XTR-1

°O ['((v + 2k + p + l)/2) x l!F ((v + 2k + p)/2)

v + xTR-lx

5.5 Arellano-Valle and Bolfarine's Generalized t Distribution Arellano-Valle and Bolfarine (1995) considered what is being referred to as a generalized t distribution within the class of elliptical distributions. The distribution is defined by

X = µ + V1/2Y,

(5.5)

where V has the inverse gamma distribution given by the pdf (v/2)+1

v/2

f(v) _

\v)

(r(vl2)

(_A exp

2

)

'

v>0

and Y is distributed independently of V according to a p-dimensional normal distribution with mean vector 0 and covariance matrix R. We shall write X - tp(µ, R; A, v). When A = v, this distribution reduces to the usual multivariate t distribution (1.1) with mean vector it, correlation matrix R, and degrees of freedom v. For R > 0, the joint pdf of

X -tp(µ,R;A,v) is f (x)

_

[1+(X_)TR1(X_)]

F ((v + p)l2)

(v+p)l2

RI12

E (X) = µ,

v>1

(5.7)

and

Var (X)

=

v

A

2R,

v > 2.

Furthermore, for an m x 1 vector q and an m x p matrix B,

Z = 77+BX

(5.8)

5.5 Arellano-Valle and Bolfarine's Generalized t Distribution

(77+Bµ)+V1/2BY t,,, (i + Bµ, BRBT; ,\ v)

95

(5.9)

since BY has the m-dimensional normal distribution with mean vector 0 and correlation matrix BRBT. Now let

X=

/ 1

X2

I

(5.10)

,

(5.11) and

R =

R11

R12

R21

R22

(5.12)

where Xl is m x 1, R11 is m x m and so on. Taking B = [I,,,,, 0] in (5.9),

note that Xl - tn,,(pl, R, 1; A, v). By symmetry, X2 - tp_m(µ2i R22; A, v). Assuming R > 0, let = µl + R12R22 (x2 - A2),

Al (x2)

R11 - R12R22 R21,

R112 and

q (x2)

=

(x2 - µ2)T R22 (x2 - A2)

Using the fact that IRI

=

and

(x - µ)T R-1 (x - µ) _ (xl - µ1)T Rl11.2 (x2 - µ1 (x2)) + q (x2) note that the conditional pdf of Xl given X2 = x2 is given by

f (xl

I X2)

_

,

)}("+p-m)/2 F ((v + p)/2) {A + q (x2 A + q (x2) 7rm/2f ((v + p - m)/2) IRI1/2

-(v+p)/2

+ (x1 - Al

(X2))T

R11'.2 (xl

- µl (x2))

This means that X1 I X2 =x2 ^'

tm(µ1 (x2),R112;A+q(x2),v+p-m). (5.13)

96

Multivariate Generalizations and Related Distributions

Note that when.\= v, Xi I X2 = x2 - tm(µi (x2) , R11.2; v+q (x2) , v+ p - m). Since q(x2) 34 p - m, this shows that the usual t distribution does not retain its conditional distributions (see Section 1.11). Finally, it follows from (5.13) and (5.7)-(5.8) that

=

E (X1 I X2)

Jz1

+ R11R22 (X2 - 1i2)

and

0027 (X 1 I X 2 )

_

+ (X2 -µ2)T R22 (X2 - A2)

v+p-m-2

x (R11 - R12R22 R21) .

Arellano-Valle and Bolfarine (1995) also presented characterizations of the generalized t distribution (5.5) in terms of marginal distributions, conditional distributions, quadratic forms, and within the class of compound normal distributions. Briefly, these characterizations are Let X have the p-variate elliptically symmetric distribution with mean vector It and covariance matrix R (for a definition of an elliptically symmetric distribution see, for example, Fang et al., 1990). Then, any marginal distribution is a generalized t distribution if and only if X has a generalized t distribution. Let X = (XT,, XZ )T have the p-variate elliptically symmetric distri-

bution with mean vector µ and covariance matrix R, where X1 is m x 1. Then, the conditional distribution of X1 given X2 is the generalized m-variate t distribution if and only if the distribution of X is the generalized p-variate t distribution. The proof of this result, which assumes the existence of a density, is similar to the proof considered in the pioneering paper by Kelker (1970) for the characterization of the multivariate normal distribution. Let X have the p-variate elliptically symmetric distribution with mean vector 0 and covariance matrix Ip, and let A be a symmetric p x p matrix. Then, XT AX - (mA/v)F,,,,,u if and only if X - tp(0, Ip; A, v),

A2 = A, and rank(A) = in. This result is proved by utilizing Anderson and Fang's (1987) assertion on the spherical distributions that put zero mass in the origin. The fourth characterization within the class of compound normal distributions is a consequence of a well known theorem due to Diaconis and Ylvisaker (1979), which asserts that, in the regular exponential family

with the natural parameterization, if the posterior expectation is linear, then the prior distribution must be conjugated. It states that if

5.6 Fang et al.'s Asymmetric Multivariate t Distribution

97

X1, X2, ... is an infinite sequence of orthogonally invariant random vari-

ables (which means that for each p, X = (X1,. .. , XP)T and I'X are identically distributed, for all p x p orthogonal matrices r) such that X1 = 0 with probability zero and Var(X2 I X1)

= b + aXi,

0 < a < 1,

b > 0,

(5.14)

then X is distributed as tp(0, Ip, b/a, (a+1)/a). The converse also holds. Arellano-Valle et al. (1994) pointed out that (5.14) could be extended to yield a location mixture of generalized t distributions as follows. Let X1, X2, ... be an infinite sequence random variables such that for each p, X = (X1, ... , Xp)T and FX are identically distributed, for all p x p orthogonal matrices I' satisfying rip = ip (where lp is a p-dimensional vector of 1's). Under this assumption there exists random variables M and V such that, conditional on M and V, X1i X2, ... are independent and normally distributed with mean M and variance V. Actually, M and V can be interpreted as the limits n 1

Xn = n

Xi

-4 M

i=1

and

(Xi-Xn)2 sn= 1n i-1

-- V

as n --* oo, where the convergence is with probability 1. Furthermore, if

Var {(X2 - M)2 I X1, M} = a (Xi - M)2 + b,

00, (5.15) then X is a location mixture of tp(Mlp, Ip; b/a, (a + 1)/a) and, in addition, M and V are independent. Because of the form of the conditions (5.14)-(5.15), these two results are known as the predictivistic characterizations of the generalized t distribution. These results could be extended further to the matrix-variate t distributions (see Section 5.11).

5.6 Fang et al.'s Asymmetric Multivariate t Distribution Fang et al. (2002) introduced an asymmetric p-variate t distribution with degrees of freedom (m, mi, ... , mp). Its joint pdf is given by .f (x)

=

Y'

(Tml (Tm, (xi )) , ... ,

T ' (Tm,,

p

(x1)) i R) [J tm; (xi) , i=1

98

Multivariate Generalizations and Related Distributions (5.16)

where

0(y1,...,yP;R)

_

r((m+p)/2)rP-1 (m/2) 11+ YTR 1Y I- MT+Z FP ((m + 1)/2) JR1 P

\

//

2 V\ (m+l)/2

i=1

Here, R denotes the correlation matrix, and tm and T,,,, respectively, denote the pdf and the cdf of the Student's t distribution with degrees of freedom m. Note that the marginals of (5.16) have different degrees of freedom. In the particular case mi = m, i = 1, . . . , p, (5.16) reduces to the usual p-variate t distribution with degrees of freedom m.

5.7 Gupta's Skewed Multivariate t Distribution In the next four sections (starting with this section), we shall discuss skewed multivariate t distributions - a topic that has received special attention in the last few years, following the introduction of the skewed multivariate normal distribution in the classical paper by Azzalini and Dalla Valle (1996). A careful reader will observe that the possibilities of constructing skewed multivariate t distributions are practically limitless. A p-variate random vector Y = (Y1, Y2, . . . , YP)T is said to have the skewed normal distribution if its joint pdf is given by

fy(y) = 20p (y; E),P(aTY) ,

Y E RP,

(5.17)

where E > 0 (with R denoting the corresponding correlation matrix), a E RP, Op(y; E) is the p-dimensional normal density with zero means and covariance matrix E, and is the cdf of the standard normal distribution. Let W be a chi-squared random variable with degrees of freedom v, distributed independently of Y. Gupta (2000) defined the joint distribution of

Xi =

W/v ,

j = 1, 2, ... , p

(5.18)

as the skewed multivariate t distribution with degrees of freedom v. The joint pdf of (5.18) is given by T

fx(x; a) = 2fv(x)Fv+P

v

ax

v+XTE 1X

(5.19)

5.7 Gupta's Skewed Multivariate t Distribution (b)

(a)

-4

-2

1

-2

0

0

2

4

(d)

(C)

-4

99

2

4

-4

-2

I

0

1

2

1

4

Fig. 5.1. Fang et al.'s asymmetric t pdf (5.16) in the bivariate case (a) m = 2, m1 = 10, m2 = 10, and r12 = 0; (b) m = 2, ml = 10, m2 = 2, and r12 = 0; (c)

m=2,m1=10,m2=10,andr12=0.5,and(d)m=2,ml=10,m2=10, and r12 = 0.9

where x E RP. Here, fk and Fk, respectively, denote the joint pdf of the central p-variate t distribution with correlation matrix R and degrees of freedom k and the cdf of the Student's t distribution with degrees of freedom k. From the definition (5.18) and the joint pdf (5.19), Gupta noted the following properties

If a = 0, then (5.19) reduces to the central p-variate t distribution with correlation matrix R and degrees of freedom v. The skewed multivariate t distribution approaches the skewed multivariate normal distribution as v -4 oo, that is,

lim fx(x; a) = 2cb (x; E) (D (aT x)

100

Multivariate Generalizations and Related Distributions

Since Yj2 is a chi-squared random variable with degree of freedom 1, Y2

X=

- F(1, v),

Wj

the F distribution with degrees of freedom 1 and v; furthermore, the joint distribution of (X 1,X2 , ... , is multivariate F with parame ters 1, 1, ... , 1, v + p Since yT E-1y is a chi-squared random variable with degrees of freedom p, the quadratic form 1

xE-1x - YE-y W/V

pFp,,,;

note that - as in the case for multivariate normal - the distribution of this quadratic form does not depend on a.

The special case of (5.19) for E = Ip is called the standard skewed multivariate t distribution. If, in addition, v = 1, then it is defined as the skewed multivariate Cauchy distribution with the joint pdf -(p+1)/2

fx(x;a) _

(p

27r-(p+1)/2I

11

2 J

0

(1+x)2 j=1

p+lax

xFp+1

xERp.

E1'=1x It should be noted that the above does not belong to the class of elliptically symmetric distributions, whereas the multivariate Cauchy does. Using results in Azzalini and Dalla Valle (1996) and Gupta and Kollo (2000), the mean vector and the covariance matrix associated with (5.19) are calculated as

E (X) =

Ea

2v

(v - 2) 1 +aTEa' v > 2,

and (

Cov X

)

=

v

r

(v-2)(v-4)E[1 v>4,

2(v + 4)aaT E

ir(v-2)(1+aTEa)

1

'

5.7 Gupta's Skewed Multivariate t Distribution

101

respectively. Furthermore, using the definition (5.18), the product moments are easily obtained as µrl,r2i...,rp

=E

11

j=1 P

vr/2E (W-r) E II Yjr' j=1

_

v r ((v/2) - r) E r(v12)

2

for r < v/2, where r = r1 + r2 +

ft 11 yri

(j=1

+ rP. If Y1, Y2,. . ., YP are mutually

independent, then the right-hand side can be easily calculated. Branco and Dey (2001) noted that the joint pdf (5.19) is a particular case of a general class of skewed multivariate elliptical distributions. Actually, the joint pdf of the general class takes the form

fx (x)

=

ff,T (x) Fv*,T.

(AT (x

- µ))

(5.20)

where v* = v + p, T*

= T + (X - µ)T R-1 (X - Fl) , QTR-1 aaTR-1Q'

1

=

fv,T (x)

r ((v + p)/2)

f1 +

(x - µ)T R-1 (x - µ)1 v

(7rA)P/2r(v/2) (RI1/2 L

1

and

Fv' T*

(x)

(7-*), */2

=

r ((v* +1)/2)

fX

VITrr (v*/2)

(T* + y2)

dy.

00

Note that f v,, (x) is the generalized t pdf described in equation (5.6), and that Fv*,T.(x) is the cdf of a generalized version of the Student's t distribution. The mean and the variance of the univariate marginals of (5.20) are

E (X) = ar ((v - 1)/2) IT r (v/2)

7r

Multivariate Generalizations and Related Distributions

102

(provided v > 1) and

Var (X) =

a2 v

V

[.((v - 1)/2)12

v - 2

IF (v12)

1

(provided v > 2), respectively.

5.8 Sahu et al.'s Skewed Multivariate t Distribution Using transformation and conditioning, Sahu et al. (2000) obtained a skewed multivariate t distribution given by the joint pdf

r_V+ _P

2ta,v (x; IL, R + D2) Tin,v+,,, x

v + q(Y)

12

(I - D (R + D2) -1 D)

D (R + D2) -1

where y = x - µ, q(y) = y' (R + D2)-ly, and D is a diagonal matrix with the skewness parameters S1, ..., 5,,,,. In (5.21), tm,v(tt,1) denotes the usual m-variate t density with mean vector µ, correlation matrix 0, and degrees of freedom v. Furthermore, denotes the joint cdf of t,,,,,v(0,I). The mean and the variance of this skewed t distribution are given by

1)/ 2)

((V

E(X) = N + V

r (v/)

and

Cov(X)

=

v v

2

(R + D2)

- ii

I

F

((v(1)/2) 12 D2 F v12)

(provided v > 2), respectively, where S = (Sl, ... ,

,,.)T . The multivariate skewness measure /31,,,,, (Mardia, 1970b) can be calculated in analytic

form. The expression does not simplify and involves nonlinear interactions between the degrees of freedom (v) and the skewness parameter S when D = SI. However, /31,,,,, approaches ±1 as S -> ±oo. Sahu et al. (2000) discussed an application of this model in Bayesian regression models.

5.9 Azzalini and Capitanio's Skewed Multivariate t Distribution103

5.9 Azzalini and Capitanio's Skewed Multivariate t Distribution A slight extension of the skewed normal distribution given in (5.17) is

fy (y) _

(aTW-1(y-4)),

(5.22)

where y E RP, E E RP, W = diag( -u 1, ... , -a ), and the rest is as defined in (5.17). In the particular case = 0, (5.22) reduces to (5.17). Starting with a random vector Y having the pdf (5.22) with = 0, Azzalini and Capitanio (2002) defined a skewed t variate as the scale mixture

X=

+ Y/ /VV,

(5.23)

where vV is distributed independently of Y according to a chi-squared distribution with degrees of freedom v. Simple calculations using a preliminary result on Gamma variates show that the joint pdf of X is

(aTW_1 (x

fx(x) = 2f (x)

-)

T +_V

where Q = (x - )TR-1(x - ) and fk, Fk are as defined in (5.19). Note that this pdf coincides with that of Branco and Dey (2001) given in (5.20). In the standard case = 0 and E = R, the joint cdf of X can be represented as

Fx(x) = 2 Pr (-Uo/-,/V < 0, U/VV < x) ,

(5.24)

where (Uo, UT)T has the (p + 1)-dimensional normal distribution with zero means and covariance matrix given by 1

(S

ST

R

where

Ra 1

aaTRa

The representation (5.24) can also be written in terms of a (p + 1)dimensional t distribution. In the case = 0, simple expressions for the moments of X can be obtained. Defining Fi

F ((v - 1)/2) 8, it

F (v/2)

104

Multivariate Generalizations and Related Distributions

and provided that v > 1, one obtains

= Wkklik,

E (Xk)

E(X2) _ v V- 2wkk, 2

v (3 -STS) 3 E (Xk)

E Xk4)

v-3

3

wkkµk,

3v2

4

(v - 2) (v - 4) wkk,

=

E (X) = Wµ, E (XXT) =

Skewness (Xk) =

E, vv 2

v (3 - (ST(S)

v-3

Ilk

v> 2,

3v

_

3/2

V

v-2

+ 2µTµ

v-2

v>3,

T

and Kurtosis (Xk)

3v2 f (v-2)(v-4)

+6vµT v-2µ

4vµT µ (3

- 6T6)

v-3

- 3 (,5T'5)2 /

v

v-2

- µT/-6

2'

v>4. Properties concerning linear and quadratic forms of X can also be derived. For example, if a E R'' and A is a m x p constant matrix of rank m, then the affine transformation a + AX will also follow the skewed t distribution given by (5.23) with the parameters , E, and a replaced by a + E', and a', respectively (the degrees of freedom v remains unchanged), where

E'

= AEA,T

5.10 Jones' Skewed Multivariate t Distribution

W'

a

105

BTa

1 + aT (E" - B (V)_1 BT) a

W' = E', B = W-TEAT, and E" is given by E = WE"W. Also, for appropriate choices of B (a symmetric p x p matrix), the quadratic form Q = (X - ) TB (X - ) can be shown to have the f Ff,,, distribution for some degrees of freedom f. For details see Azzalini and Capitanio (1999) and and Capitanio et al. (2002). A further extension of (5.17) examined independently by Arnold and Beaver (2000) and Capitanio et al. (2002) is of the form

fy (y) = Op (y - ; E),t (ao + aT W-1 (y

- )) /41(T), (5.25)

where y E llw', T E R ao = T/ 1 - 6TR-16, and the rest are as defined in (5.22). In the particular case T = 0, (5.25) reduces to (5.22). Taking Y in (5.23) to have the pdf (5.25) with = 0, one obtains an extended skewed t distribution for X. The corresponding joint pdf for X is quite complicated, but the joint cdf can be represented as

F(x)

Pr(-(Uo+T)/V/V-
=

(compare with (5.24)). Moreover, for the particular case and second-order moments are

E (X)

= 0, the first-

= E (1/VV) rll(T)WS

(provided v > 1) and

E(XXT) =

y

+ ,(T)}W8(WoT)T] IE+f772(7-)I

v2

(provided v > 2), respectively, where 77(x)

=

dxk

for k = 0,1, 2,....

5.10 Jones' Skewed Multivariate t Distribution The univariate Student's t distribution has the pdf 1+

F((v + 1)/2) v7rr(v/2)

{

x21-(-+l)/2. v

(5.26)

106

Multivariate Generalizations and Related Distributions

By replacing (5.26) with the skewed univariate t pdf (4.38) in a multivariate distribution, Jones (2002c) introduced a new skewed multivariate t distribution that we shall describe in this section. Let X be a p-variate random vector having the standard multivariate t distribution with the joint pdf given by

r((v + p)/2)

1 + xTx v

(v+p)/2

(5.27)

}

The univariate marginals of this are (5.26). Multiplying (5.27) by (4.38) and dividing by (5.26) yields Jones' (2002c) skewed multivariate t distribution. The corresponding joint pdf is

21-a-cr((y + p)/2)r(a + c) (v7r)(P-1)/2

a + cr((v + 1)/2)r(a)r(c) x2

(

x{1+ 1} v

a+1/2

(v+1)/2

1+

ll

X

1-

x1

a+c+xl xTx -(v+p)/2

c+1/2 (

X1

{1+`-}

a+c+x1

.

(5.28)

v J

111

This reduces to (5.27) for a = c = v/2. In the bivariate case, (5.28) is a distribution with (i) a skewed t marginal with parameters a and c in the x1 direction; (ii) conditional distributions of X2 I X1 that match those of the bivariate t distribution being t distributions on v + 1 degrees of freedom scaled by a factor of (v -x2,)/(v + 1); and (iii) a diagonal correlation matrix. Another new multivariate distribution can be obtained by replacing (5.26) by the pdf of the Gumbel distribution: exp(-x1 - exp(-x1)). This results in the joint pdf r ((v + p)/2) (v7r)(p-1)/2r ((v + 1)/2) X (1

x2

+ 1V I

exp {-x1 - exp (-x1)} xTx -(v+p)/2

(v+1)/2

{1+

l

v

}

(5.29)

With respect to the correlation structure, this pdf has much in common with (5.28). But the conditional distribution of X1 given X2, ... , Xp and the marginals are different. Jones (2001a) noted that, if Y has the beta distribution with parameters a and c, then X = a + cY/ 1 - Y2 has the skewed univariate t distribution given in (4.38). Jones (2002c) observed a similar relation-

5.10 Jones' Skewed Multivariate t Distribution

107

(b)

(a)

Nx

-4 -2

0

2

-4 -2

4

0

2

4

X1

X1

Fig. 5.2. Jones' skewed multivariate t pdf (5.28) for p = 2 and (a) a = 6,

v=3,andc=2; and (b)a=2,v=3,andc=6 ship between the joint beta pdf

21-"-`r(a + c)F(b)

7rr(a)r(b - 1/2)I (c)

b

(1 + y1)"

(1 -

y1)1-b (1

- yi - Y2) 6-1

a>0, b>1/2, c>0 and the skewed multivariate t distribution given in (5.28) when p = 2 and b = v/2 + 1; namely, if (Y1i Y2) have the former distribution, then (X

1

7

2

)

_

(VYi a+c 1-Y12

Y2 VV +Y2(a+c-v)

1-Y2-YZ 1-Y2

has the distribution (5.28) for p = 2. In the univariate case, F and skewed t (equation (4.38)) distributions are linked in two ways that produce identical results: (i) A random

108

Multivariate Generalizations and Related Distributions

(a)

(b)

NX

NX

-2

0

2

-2

4

2

0

X1

4

X1

Fig. 5.3. Jones' skewed multivariate t pdf (5.29) for p = 2 and (a) v = 1; and (b) v = 20

variable with any one distribution can be obtained by transforming a random variable from the other; (ii) a random variable with each distribution can be written as a function of two independent chi-squared random variables. If Wi ti X22,,,, Fi ti F2ii,,,,,, and Ti is a random variable

with the pdf (4.38), then Ti

2

= F=

=

Ti

=

l V0

wivi

v2

Fi -

vo

vi vo

-/

(T,

Fi 2

+

wi+T,?

(5.30)

5.10 Jones' Skewed Multivariate t Distribution

109

and

vo Wi

=

Fi

Vi WO,

where wi = vo + vi. By extending this relationship between the univari-

ate F and the skewed univariate t, Jones (2001b) introduced another skewed multivariate t distribution. It is known (see, for example, Johnson and Kotz, 1972, Chapter 40, and Hutchinson and Lai, 1990, Section

6.3) that the joint pdf of the random variables Fi, i = 1,.. . , p is vkk

r(n) r(vo)...r(VP)

f (fl, ... I M =

k=0

fj'-1

j=1

r

1v0+Ej=lvjfj)n

fi>0,...,fP>0,

(5.31)

where n = vo + + vP. Applying the transformation (5.30) to (5.31), Jones (2001b) obtained the joint pdf of Ti as f (t1, ... , tP)

=

F(n) I (v0) ... F (VP)

2 (tk +

P

wvk k

k=0

1+ P

2vk

Wk + tk)

wk + t2

(tj+ wj+ 2

2

n

Wi

j=1

ti E R,...tp E R.

(5.32)

The univariate marginals of this pdf take the form of (4.38). The conditional pdf of Ti given any subset Ti, , ... , Ti,,, of the other variables, P2 < p, is proportional to

(ti + wi + ti )

2v;

wi+ti (1+K-1 (ti+ wi+ti)

2

W1 +vi 1 +...+v'p,

where P2

K = wi

1+

2

we

(ti, +

w + ti, )

l=1

The regression of Ti given T1,.. . , Tip, takes the nonlinear form E (Ti

T'".

.

,Tip2)

=

r(

- 1/2) r (vi - 1/2) ((vi - 1/2) v 2r (o) r (vi)

vo

110

Multivariate Generalizations and Related Distributions vowi (0 - 1/2)

+ vip2 . Note that the corresponding relation where 0 = vo + vi, + for the multivariate F distribution in (5.31) is linear. If Ti,. .. ,T,,,, -

denote any m of the p Ti's (with their degrees of freedom correspondingly

renumbered as vi, ... , T1i...,TM is m F

TTTA+

along with vo), then the product moment of

wa,/2...WA /2

-

1

I-F(vi)...F(V-) Al A

ji= o

j_ = o m

xr i-1

r-\

2

i

i.1 m

vi

A,) C'.

A,

C1 A.

x

.7i-

+

2

+vo

,

provided that vi > Ai/2, i = 1, ... , m and vo > (A1 + .

+ Am)/2. In

particular, the variances and the covariances are given by

r,

1-1

wi l

,

1

2

x - v0) (vo

+

1) (Vi - 1)

F (yo - 1/2) F (vi - 1/2)

2

r (vo) F (vi)

wi-2 (vo - 1) (vi - 1)

(provided vo > 1 and vi > 1) and

Cov (Ti, Tj )

wiwjr (vi - 1/2) F (vj - 1/2) 4F (vi) F (vj)

1+vo-vi-vj

+ (vi - 1/2) (vj - 1/2) (vo - 1/2)

-(vi-u0)(ii -v0) (

F(vo-1/2)2 ) F (v0)

(provided vi > 1/2, vj > 1/2 and vo > 1), respectively. In the particular

5.10 Jones' Skewed Multivariate t Distribution

case vo -

111

= vP = v/2, (5.32) reduces to

f (t1 ... , tP)

=

I ((p + 1)v/2)

P

rP+1 (v/2)

V + tk {(tk+v)v}

/{V+(t+)2} P

(P+1)v/2

j=1

t1 E R'. .. , tP E

(5.33)

Jones (2001b) referred to this distribution as the symmetric multivariate t distribution. Note that all of the marginals of (5.33) have the Student's t distribution with degrees of freedom v. The correlation between any two Ti's in (5.33) takes the simple form

_ 2v-3 F((v-1)/2)) 2 P

( r (v/2)

8

provided that v > 2. The limiting form of (5.32) as vo --* oo and vi > 1 remains fixed, i = 1, ... , p can be shown to be P

110ri

9 (µ1 + Q1t1i ... , µP + °PtP)

i-1 where

9 (t1, .. , tP)

P tir (2vi+1) = 2P [J (v)

exp

(- tz

8

i=1

1

2 /

F(yi-1/2) r(vi)

ILi

and

Qi

=

1

vi-1

P2

(vi -V,)1/2) r2

Note that µi and Qi are the mean and the standard deviation of 2/x2V; distribution. When v o remains fixed but. VI, .. , vP -* oo, the marginals of (5.32) tend to distribution, but the correlations between the Ti's tend to 1 and the joint distribution becomes degenerate. When all v0i v1, ... , vP -* oo, all of the marginals tend to the normal distribution

- but the form of the limiting joint distribution will depend on the

Multivariate Generalizations and Related Distributions

112

(b)

(a)

v N X

0 N 5

10

X1

X1

(C)

(d)

X

15

X

X1

X1

Fig. 5.4. Jones' skewed multivariate t pdf (5.32) for p = 2 and (a) vo = 2, vi = 4, and v2 = 4; (b) vo = 2, vi = 20, and v2 = 1; (c) vo = 2, vi = 1, and

v2= 20; and (d)vo=v1=v2=2

specific relationships between the v's. The limit of (5.33) as v -4 00 is the multivariate normal distribution with zero means, unit variances, and an intraclass correlation structure with correlation 1/2.

5.11 Matrix-Variate t Distribution The matrix-variate t distribution, motivated by applications in Bayesian inference, is the product of James Dickey's research in the mid-1960s. We need the following terminology to discuss its mathematical properties.

Let µ be a p x q constant matrix, let R> 0 be a p x p matrix, and let

5.11 Matrix-Variate t Distribution

113

Q>0beagxgmatrix. Form>p+q-1 define k(m,p, q) =

7r111'

rg((m - p)/2)

(5.34)

rq(m/2)

where

=

I" (X)

7rr(r-1)/4r(x)r

Cx -

2) ... r Cx

+2 2

is the generalized gamma function. Furthermore, for real or complex constants a1, ... , ap and b1, ... , bq and for random matrices S and T, define the general hypergeometric functions (see Constantine, 1963) pFq (a1.... , ap; b1, ... , bq; S)

(a,),...(ap), C. (S)

co

L.

(bl), ... (bq),,

(5.35)

k!

and

pFq (a,,...,ap;bl,...,bq;S,T) 00

FE k =o ,4

(a,),, ... (a'p), C. (S) C. (T) (bi),4 ... (bq),

(5.36)

k!

where ,={k1,...,k,,,,}, k1>k2>...>km>0,k1+k2+

W. rT(x,r.)

7r

rcr9 >

+km=k,

rm (x_ n) rm(x)

/ r(x+k1)r x+k2 - 21...r (x+k,. - r 2-

and C, (S) and C,, (T) are symmetric homogeneous polynomials of degree

k in the latent roots of S and T, respectively. A p x q random matrix X is said to have the matrix-variate t distribution with parameters µ, R, Q, and m if its joint pdf is

f (X) =

IQI(m,-p)/2 IRI-q/2

k(m,1p, q)

x Q + (X - µ)T R-1 (X - µ)

-m/2 (5.37)

(Dickey, 1966a, 1967b). If µ = 0, then we say that X has the central matrix-variate t distribution with parameters R, Q, and m. Otherwise, we refer to the distribution as a noncentral matrix-variate t. The usual multivariate t distribution (1.1) is the special case of (5.37) for p = 1

114

Multivariate Generalizations and Related Distributions

(single row) or q = 1 (single column). It is also known that the partic-

ular case of (5.37) for µ = 0 and R = Ip is a mixture of the normal density with zero means and covariance matrix Ip ® V - in the q x q positive definite scale matrix V. Densities of the form (5.37) appear in the frequentist approach to normal regression as the distribution of the Studentized error, both the error in the least squares estimate of the coefficients matrix and the error in the corresponding predictor of a future data array (Cornish, 1954; Kshirsagar, 1961; Kiefer and Schwartz, 1965). In the Bayesian conjugate-prior and diffuse-prior analyses for the same sampling models, it arises as the marginal prior or posterior distribution of the unknown coefficients matrix, and also as the predictive distribution of a future data array (Geisser and Cornfield, 1963; Ando and Kaufman, 1965; Geisser, 1965; Dickey, 1967b, Section 4; Zellner, 1971, Chapter 8; Press, 1972, Section 8.6). More recently, Van Dijk (1985, 1986) discussed applications of (5.37) in the linear simultaneous equation (SEM) model, which is one of the best-known models in econometrics. The SEM model is used in several areas, for instance, in microeconomic modeling for the description of the operation of a market for a particular economic commodity and in macroeconomic modeling for the description of the interrelations between a large number of macroeconomic variables.

If X has the central matrix-variate distribution with parameters R, Q and m, then it can be represented in numerous ways, as described by Dickey (1967b) and Dawid (1981). The following results (due to Dickey, 1967b, and Dickey et al., 1986) concern the conditional and the marginal distributions of X

If X = (X1, X2)T, then the conditional distribution of X1, given X2, is the matrix-variate t with parameters -R11R12 X2, Rj11, Q + X2 R22 X2, and m. If X = (XI, X2), then the conditional distribution of X1, given X2, is a matrix-variate t with parameters X2Q22 Q21, (R+X2Q22X2)-1, Q11 - Qi2Q22Q21, and m. If X = (X1, X2)T, where Xi is pi x q, then the marginal distribution of X2 is a central matrix-variate t with parameters R22 , Q and m - pl. In the particular case XT = (xi.... , xp), each row xT has the central multivariate t distribution with degrees of freedom m - p - q + 1 and correlation matrix riiQ/(m - p - q + 1). A consequence of this is that the density (5.37) of X can be written as the product of conditional

5.11 Matrix-Variate t Distribution

115

multivariate t distributions of the rows of X, that is,

f (X) = f (XI) f (X2 I XI) ...

f (XP I X1, ... , Xp-1)

If X = (X1, X2), where Xj is p x qj, then the marginal distribution of X2 is a central matrix-variate t with parameters R-1, Q22, and m-q1. In the particular case X = ( X I ,- .. , x9 ), each column xj has the central multivariate t distribution with degrees of freedom m - p - q + 1 and correlation matrix giiR/ (m - p - q + 1). A consequence of this is that the density (5.37) of X can be written as the product of conditional multivariate t distributions of the columns of X, that is,

f (X)

=f(XI)

(X2

X1)... f (Xq I X 1 ,

I

If X is doubly partitioned as

X=

X11

X12

X21

X22

'

where X ij is pi x qj with p1 + P2 = p and q1 + q2 = q, then the conditional distribution of X12 given X11 and X21 is a matrix-variate t with parameters R112 11 -1X11, Q11 + XT R1-1'X11, R22 - R21R111R12, and m + q1 - p- q + 1. (Here, the partitions of R and Q correspond to the partition of X.). Since this depends only on X11, it follows that X12 and X21 given X11 are conditionally independent.

The following results (due to Javier and Gupta, 1985, and Dickey et al., 1986) concern the distributions of the quadratic forms XAXT and AXB when X has the central matrix-variate t distribution with parameters R, Q, and m. If A > 0 is q x q, then the pdf of W = XAXT is given by

f (W) =

IAI-P/2

1

IRI(m-q)/2

k(m,p,q)

x IR+r

IQI-P/2 IWI(q-P-1)/2

WI--/2

x 1Fo \ 2 . (R + W)-1 W, Iq

- (QA)-1)

where W > 0, k(m, p, q) is given by (5.34), and 1Fo is as defined in (5.35). An immediate consequence of this result is that

I

W(P1)/2 IR+ WI-m/2 O

x 1Fo \ 2 . (R

+ W)-1 W, 1q- (QA)-1) dW

116

Multivariate Generalizations and Related Distributions

= F (q/2)Ir

q)/2) IAIp/2 IRI-(m-q)/2

jQjp/2

Hence the hth moment of I W I is

_ F ((q + 2h)/2)1' ((m - q - 2h)/2)

[IWIh]

E

1' (m/2) x IAIp/2 IRI-(m-q-2h)/2 IQIp/2.

(5.38)

Further using the fact that an F-distribution is uniquely determined by its moments, it follows that I W I can be written as a product of q independent univariate F's, that is, q

IWI

^'

rIF'(q-(j-q),m-q-(j-1)) j=1

For the special case A = Ip and p = q, (5.38) gives the hth moment of XXT.

If A > 0 is p x p and B > 0 is q x q, then AXB has the central matrix-variate t distribution with parameters BTQB, AR-'A T , and

m,m>p+q-1.

If A > 0 is p x p and B is a q x r rectangular matrix, then AXB has the central matrix-variate t distribution with parameters BTQB, AR-1A T , and m, m > p + r - 1. If a is a q x 1 vector, then aT XT has the central multivariate t distribution with degrees of freedom m - p - q + 1 and correlation matrix a TQaR/(m - p - q + 1). If a is a q x 1 vector such that aTa = 1 and b is a p x 1 vector, then aT XT b is a linear combination of Student's t random variables. If b is a p x 1 vector, then XTb has the central multivariate t distribution with degrees of freedom m - p - q + 1 and correlation matrix

bTRbQ/(m-p-q+ 1). If a is a q x 1 vector and b is a p x 1 vector, then

(m - p - q + 1)a TXT b (aTQa) (bTRb) has the Student's t distribution with degrees of freedom m - p - q + 1.

In the special case R = Ip and Q = 1, if a is a real number and b is a q x 1 vector such that a2bTb = 1, then aXb has the Student's t distribution with degrees of freedom m - q. Javier and Gupta (1985) also derived a useful factorization of the central matrix-variate t density in terms of the product of q - 1 independent

5.11 Matrix-Variate t Distribution

117

univariate F densities and q independent multivariate t densities - paralleling the result of Tan (1969a) for matrix-variate beta distributions. Let X be a p x q random matrix having the density (5.37) with It = 0. Set

U=

(R-1/2XQ-1/2) (R-1/2XQ-1/2)T

so that U is p x p, symmetric, and U > 0. Partition U as

U =

CU11

U21

U12 U22

so that U11 is 1 x 1 and U22 is (p - 1) x (p - 1). Abbreviating D22 D21D111D12 by D22.1, define the following submatrices U2221

=

U22.11) 22.1

U22.1

-

U 11

=

( U22.11) 11

U22.1, j = 1 2,...,p - 1,

and

Ui°)

= U11,

so that is (p - j) x (p - j) and U11) is 1 x 1. With all of this notation the factorization of the density of X (due to Javier and Gupta) can be stated as q-2 m-jq-2(p-.7)-2 f(X) _ 11 F p-j+2 2 2 )

j=0

P

X

1I tUii) {1+uM

(j

IP-j + U22,1; m - (j - 1) q

j=1

where t (T; r) is the joint pdf of a central multivariate t distribution with degrees of freedom v and correlation matrix T and F(a, /j) is the pdf of a univariate F distribution with degrees of freedom a and /3. The two predictivistic characterizations of the multivariate t distribution based on (5.14) and (5.15) have the following matrix-variate generalizations Let X1, X2, ... be an infinite sequence of q-dimensional random column vectors that are orthogonally invariant (which means that, for

118

Multivariate Generalizations and Related Distributions

each k, X(k) = (X1i ... , Xk)T and rX(k) are identically distributed,

for all k x k orthogonal matrices I') and, for k fixed, let X ik)T , i = 1, 2,.... If XI, ... , Xq are linearly independent with probability 1 and

E [X(q)TX2(q) I Xiq)1

= aX(q) TX(q) + B

where 0 < a < 1 and B is a q x q positive definite matrix, then the distribution of X(P) is the matrix-variate t with µ = 0, R = IP,

Q=(1/a)B,andm=1+(p/a)-p. Let X1, X2, ... be an infinite sequence of q-dimensional random col-

umn vectors such that, for each p, X(P) and rX(P) are identically distributed, for all p x p orthogonal matrices r satisfying rip = 1P (where 1P is a p-dimensional vector of l's). Under this assumption there exists a u-algebra T of events such that n

1 Xi Xn = -1: n i=1

E(Xi IT)=M and

n

Sn

=

n i-1

(Xi - Xn) (Xi - Xn)T

E (XIXT I T) - E (XI I T) {E (XT I T) }T = V as n -* oo (Chow and Teicher, 1978), where the convergence is almost everywhere. Moreover, if

E [(xv - 1gMT)T (X(q) - 1g MT) I X(q) M] a (X(q)

- 1gMT)T (X(q) - 1gMT) + B, q\\

where 0 < a < and B is a q x symmetric positive definite matrix, then X(P) is a location mixture of the matrix-variate t distribution 1\1

with µ = 1TM, R = Ii,, Q = (1/a)B, and in = 1 + (p/a) - p. In addition, M and V are independent. Dawid (1981) provided a different but more convenient parameterization of (5.37). If Y (p x p) has the standard matrix inverse Wishart distribution with parameter 8 and if, given Y, X (n x p) has the matrix normal distribution with parameters In and Y, then X is termed

5.12 Complex Multivariate t Distribution

119

as having the standard matrix t distribution. In the notation of (5.37), this would correspond top = 0, R = I, , Q = In, and m = 6 + n + p - 1. Under Dawid's parameterization, if X* is a n* x p* submatrix of X, then X * has the matrix t distribution with parameters In-, Ir. , and S: Note that 6 is unchanged. This kind of consistency enabled Dawid (1981) to construct what is termed as the standard infinite matrix t distribution. Namely, X = {x2j, i > 1, j > 1} is said to have the above-named distribution if it has the property that for all (n, p) the leading n x p submatrix of X has the standard matrix t distribution with parameter 6. The standard matrix t distribution also has the attractive property of being spherical, that is, if P (n x n) and Q (p x p) are two orthogonal matrices, then both PX and XQ have the same distribution as X.

5.12 Complex Multivariate t Distribution A complex normal random vector Y = V +W is a complex random variable whose real and imaginary parts possess the bivariate normal distribution. A complex p-variate normal random vector

Y = v+Tw (Vl+

Wl, V2+ JW2i...,VP+

WT (5.39)

is a p-tuple of complex normal random vectors such that the vector of real and imaginary parts (Vi, Wl,... , Vr, Wr) has the 2p-variate normal distribution (Goodman, 1963). Section 45.13 of Kotz et al. (2000) provides an account of this distribution. It is usually assumed that the 2p-variate normal distribution of (V1, Wl,... , VP, Wr) has zero means and covariance matrix given by

1 (E1 -E2 1

2 ` E2

E1

where E1 is symmetric (matrix A is symmetric if AT = A) and E2 is skew-symmetric (matrix A is skew-symmetric if A = -AT) . From the given structure it is easily seen that the covariances of the p-variate vectors V and W are each equal to E1/2 and the covariance between V and W is equal to E2/2. Hence the covariance of the complex p-variate normal random vector Y in (5.39) is E1 + V/-IE2 = E, say. The properties of the distribution of Y have been studied by many authors. The joint pdf of Y is given by

f (Y) =

7rp IEI

eXP {-yTE-ly} ,

(5.40)

120

Multivariate Generalizations and Related Distributions

where y denotes the complex conjugate of y (Goodman, 1963). For ex-

ample, for the complex univariate normal distribution, y = vl +wi and the covariance matrix E = a2, and thus the joint pdf of Y becomes

vi +wi

1

a2

The characteristic function of Y can be shown to be

E [exp {i (sTV + tTW) }] = exp { - uTEu}

where u = s +t (Wooding, 1956). Explicit expressions for the moments of Y have been derived by Sultan and Tracy (1996). The complex multivariate normal distribution has applications in describing the statistical variability of estimators for the spectral density matrix of a multiple stationary normal time series and in describing the statistical variability of estimators for functions of the elements of a spectral density matrix of a multiple stationary normal time series. Relatively few results are available that deal with complex multivariate t distributions. Originally, the complex multivariate t distribution was introduced by Gupta (1964). Let Y have the complex p-variate normal distribution with zero means, common variance Q2, and covariance matrix o, 2R. Let 2vS2/Q2 have the chi-squared distribution with degrees of freedom 2v, distributed independently of Y. Then X = Y/S is said to have the complex p-variate t distribution with degrees of freedom v and correlation matrix R. By writing down the joint distribution of S and X and then integrating out S, the pdf of X can be obtained as ( v+P)

f (x)

_

IIRIP

{1+

1XTR-lx }

Tan (1973) discussed some properties of this distribution. Tan (1969b) provided a brief discussion of a complex analog of the matrix-variate t distribution given by (5.37).

5.13 Steyn's Nonnormal Distributions Strictly speaking, this section does not deal with multivariate t distributions per se. This section is about nonnormal distributions arising from the class of multivariate elliptical distributions that contains the multivariate t as a particular case. One weakness of the class of multivariate elliptical distributions is that

5.13 Steyn's Nonnormal Distributions

121

all fourth-order cumulants are expressed in terms of a single kurtosis parameter (moreover, the univariate marginals have zero skewness and the same kurtosis). In fact, the cumulant generating function (cgf) and the moment generating function (mgf) of a p-variate elliptical distribution with zero means and correlation matrix..R are K (t1,...,tp) 1

(tTRt) + 2 r { 2 (tTRt) } + >2 Ak (tT Rt) k l

(5.41)

k> 3

and M (tl,...,tp) /

(tTRt) 2

tTRt l

exp

2

I

/

\\\

1+2

1!

Bk (tT Rt) k

+

(5.42)

,

k>3

respectively, where Ak, Bk are constants and n is the kurtosis parameter. Steyn (1993) attempted to introduce meaningful multivariate distribu-

tions that are related to the elliptical distributions and that contain more than one kurtosis parameter. As an example, consider a random vector (X1, X2, X3) possessing the three-dimensional normal distribution with the mgf M (t1, t2, t3)

(t1 + t2 + t3 + 2r12t1t2 + 2r13t1t3 + 2r23t2t3)

exp

.

1

2

(5.43)

Suppose this model is placed in a changing environment that favors a change in one of the random variables, say X1, in such a way that the kurtosis should be taken into consideration. Specifically, assume that the marginal distribution of X1 is elliptical with the kurtosis parameter i1, while the conditional distribution of (X2, X3) given Xl = x1 remains unchanged. Note that (5.43) can be written as M (tl, t2, tr3) 1 2=

exp L1

J

ll L(1-ri2)t2+(1-ri3)t3+2(r23 -r12r13)t1t2}1

LLLL 2

/ XJ

2 l r exp S - 2 + (ti + r12t2 + r13t3) xl y dxl. ll

(5.44)

JJJ

Changing the probability element in the integrand in (5.44) to that of

122

Multivariate Generalizations and Related Distributions

the elliptical distribution in (5.42), one can show that the mgf changes to

= M (tl, t2, t3) exp 1 g (t1 + r12t2 + r13t3)4 + ...1

M1 (t1, t2, t3)

.

The corresponding cgf becomes K1 (t1, t2, t3)

2 (1 + t2 + t3 + 2r12t1t2 + 2r13t1t3 + 2r23t2t3)

+ 1 K1 (tl + r12t2 + r13t3)4 +

(5.45)

Setting t2 = t3 = 0, the cgf of the marginal distribution of X1 is given by

K1 (t1,

0,0)

=

2t1 +

.. .

gr.lt1

which shows (as it should) that the marginal distribution of Xl is elliptical with kurtosis parameter is (compare with equation (5.41)). However, for tl = t3 = 0 and tl = t2 = 0, one obtains Kl (0, t2, 0)

K1 (0, 0, t3)

=

=

2t22 + 18K1

(r12t2)4 + .. .

2t3 + 8n1 (r13t3)4 +

thus, the marginal distributions of X2 and X3 are also elliptical but with kurtosis parameters i1r12 and rjr13, respectively. Furthermore, for ti = 0, K1 (0,t2,t3) = 2 (t2 + 2r23t2t3 + t3) + 81

r13t3)4 + ...

(r12t2 1

which shows that the joint marginal distribution of (X2, X3) is not elliptical. The fourth-order cumulants are easily obtained from (5.45) as K2Jk = 3K1r12r13, where i + j + k = 4. Suppose now that the model given by (5.43) is placed in an environ-

ment that favors a change in not only Xl but also influences (X2, X3). Assume - in particular - that the conditional distributions of X2 given Xl = xl and X3 given (X1,X2) = (XI, X2) are elliptical with kurtosis parameters K2 and ic3, respectively. Then calculations similar to those above show that the mgf (5.43) changes to 2

M2 (tl, t2, t3)

= M (tl, tz, t3) exp

2

2

z 2

2

2

2

1

I

5.13 Steyn's Nonnormal Distributions

/2 +2r13

\2

2

t3 2.12 )

I

123

+...

(5.46)

where u1

u2

=

tl + r12t2 + r13t3,

= t2 + r23 - r12r13 t3, 1 - rig 2

Q2.1

=

2

1 - T12,

and 2

0'3.12

1-r132

(r23 - r12r13)2

1-r212

It is easily seen that the marginal distributions of X1, X2, and X3 are elliptical with kurtosis parameters given by n1, r12r.1 + o"2.1K2, and rl3r'1 + (Q3.1 - a3.12)t3, where Q3.1 = 1-r13. This time, the fourth-order cumulants are given by ICzj/

= 3 t IGlri2r13 + K20`2.,

r12r13)kI

(r23 -

,

where i + j + k = 4. In the case of '004, should be added. XP)T has Similar constructions can be performed when X = (X1' a p-variate normal distribution with zero means, covariance matrix R, and the corresponding mgf

= exp 12 tTRt)

M2 (t1, ... , tp)

.

(5.47)

Consider two environments similar to those considered above for the trivariate normal model. First, divide X into two random vectors X(1) _ (X1, ... , Xh)T and X(2) = (Xh+1, ... , XP)T , and let

R = C

R11

R12 )

R12

R22

be the corresponding partition of the correlation matrix. Also let t(l) = ( t 1 ,- .. , th)T and t(2) = (th+1 i ... , tP)T be the corresponding partition

of t. Now assume that the marginal distribution of X(1) is changed to an h-dimensional elliptical distribution with kurtosis parameter k

Multivariate Generalizations and Related Distributions

124

(a)

(b)

Nx

Nx

-3

-1

X1

X1

(C)

(d)

V

1

3

1

3

V

-3 X1

-1

X1

Fig. 5.5. Steyn's bivariate pdf corresponding to (5.46) for t3 = 0 and (a) rcl = 0.8, K2 = -0.4, and r12 = 0.2; (b) i1 = 0.8, ice = -0.4, and r12 = 0.8; (c) Kl = -0.4, K2 = 0.8, and r12 = 0.2; and (d) icl = -0.4, K2 = 0.8, and r12 = 0.8

and that the conditional distribution of X(z) given X(1) = x(1) remains unchanged. Then calculations show that (5.47) changes to M1(t)

= M(t) exp

T

(t(1) 8

+ R111R12t(z)/ z

xR11 (t(1) + R111R12t(2))

+.

.

(5.48)

Clearly, z

M1 (t(1), 0)

= exp {t(1)TRllt(1) + 8'c1 (t(1)TR11t(1)) +...

5.13 Steyn's Nonnormal Distributions

125

and

M1

(p

t(2))

= exp

2it(2)TR22t(2) 2

+1K, (t(2)TR22R111R12t(2) )

+...

which shows that the marginal distribution of X(1) is an h-dimensional elliptical distribution (as it should) while that of X(1) is not elliptical. The second-order cumulants of (5.48) are the same as those for (5.47). For the second construction, partition X2 into X(3) = (Xh,... Xh+s)T and X(4) = (Xh+s+1, ... , Xr)T, and let t(2) be partitioned correspondingly into t(3) and t(4). Let C denote the conditional covariance matrix of X2 given X1 = x1, that is, )

C = R22 - R.21R111R.12,

and let C be partitioned as

C= C

C11

C12

C12

C22

so that C11 is s x s, C12 is s x (p-h-s), and C22 is (p-h-s) x (p-h-s). Now, assuming that the distribution of X1 is elliptical with kurtosis parameter i1 and that of X(3) - E(X(3) I X(1) = x(1)) is elliptical with kurtosis parameter KK2, one can show that the mgf (5.47) changes to

M2(t)

= M(t) exp [({ (t(1) + R111R12t(2) )

T

2

xR11 (t(1) + R111R12t(2)) I +

+ 8k2 (t(3) +

T

C_11C12t(4)

) 2

xC11 (t(3) + C11C12t(4))

+ ... (5.49)

This defines the mgf of a multivariate distribution that is equal to the product of the mgf of the multivariate normal and a function of two quadratic forms in t depending on the two kurtosis parameters iGi, i =

126

Multivariate Generalizations and Related Distributions

1, 2, and on the elements of the normal covariance matrix. Setting t(2) = 0 into (5.49), we see that X(1) has an h-dimensional elliptical distribution with zero means, covariance matrix R11, and kurtosis parameter Ic1 (as

it should). If either t(1) = 0 or t(1) = 0 and t(3) = 0, then M2(t) becomes a function of three different forms.

5.14 Inverted Dirichlet Distribution There is a close connection between the multivariate t distribution defined by (1.1) and the inverted Dirichlet distribution (Cornish, 1954; Dunnett and Sobel, 1954). To see this, consider the central p-variate t distribution with the pdf XTR'-1xl -(v+p)/2

r ((y + p)/2)

=

f (x)

(7fv)p/2r (v/2) IRI

1/2

1+

JII

V

Upon transforming to the canonical variables Z = (Z1, ..., Zp), Z = PX, where P is a p x p matrix such that pTp = R-1, it is easily seen that f (z)

=

F ((v + p)/2) 1 + zTZI (v+p)/2 (7rv)p/2r (v/2) L VJ

(5.50)

In (5.50) now perform a further transformation Ti = ZZ iv, which is one-to-one in each of 2p regions with the Jacobian IJI

= 2-pvpl2/

p.

Consequently, the joint pdf of TT = (T1, . . . , Tp) becomes P

f (t) =

r ((v + p)/2) t-1/2 ... t-1/2 (1 + p 1 7fp/2F (/2) i=1

(v+p)l2

E ti)

which is the inverted p-dimensional Dirichlet distribution D'(1/2, ..., 1/2; v/2); see, for example, Kotz et al. (2000, Chapter 49).

6

Probability Integrals

There has been a very substantial amount of research carried out on probability integrals of multivariate t distributions. Most of the work was done during the pre-computer era, but recently several computer programs have been written to evaluate probability integrals. Sections 6.1 to 6.7 by now may have lost some of their usefulness but are still of substantial historical interest in addition to their mathematical value. We have decided to record these results in some detail in this book in spite of the fact that some of the expressions are quite lengthy and cumbersome. Sections 6.8 to 6.13 contain more practically relevant and modern results.

6.1 Dunnett and Sobel's Probability Integrals one of the earliest results on probability integrals is that due to Dunnett and Sobel (1954). Let (X1, X2) have the central bivariate t distribution with degrees of freedom v and the equicorrelation structure r2j = p, i $ j. The corresponding bivariate pdf is

f(xi,x2) v,P) =

{'+ xl + x2 - 2pxlx2 l

1

2 1 - p2

v(1 - p2)

}

with the probability integral P (yi, y2; v, P)

=

f (xi, x2; v, P)dxldx2.

Let

x(m,yi,y2)

(Yi - PY2)2

_

)2 + (1 - PZ)(m + y(yl - Pyz

127

(v+2)/2 (6.1)

Probability Integrals

128

and let

f

(a, b)

x( m,yi,y2)

r(a

+ b) a-1 rr(b) y (1- y)

1

dy

denote the incomplete beta function. Dunnett and Sobel (1954) evaluated exact expressions for (6.2) when v takes on positive integer values. For even v and odd v, they obtained P(y1, y2; v, P)

=

1 arctan

1-P

2

-P

27r

F(j - 1/2)

+ y2 vir jj=1

4

1/2-j

+ y2 v

Cl

1

rr

x L1 + sgn(yl - PY2)Ix(v,y,,y2)

vf2 r(j - 1/2) (1 + y1

+ y1 4

v7r

x

1:

r(j)

j=1

11

1

2,7 -

2

12-j

v (21,

+ sgn(y2 - PY1)Ix(v,y2,y,)

j-2)J

and

P(y1, y2; v, P)

=

27r

±vaai }

arctan { -VvL

+

(v-1)/2

y2

4

v-7r

E j=1

+

r(j)

r(j + 1/2) (

-7

y2

v 1

x [1 + sgn(yl - PY2)Ix(v,yl,y2) (v-1)/2

+ y1 4

vir

E j=1

I(j)/ r(j+1

r

j)

1+y1-j

2)

v/

x I1+sgn(y2-Py1)Ix(v,y2,y,) L

2

G 'j) (6.4)

respectively. Here, a = y1 + y2, # = Y1Y2 + PV,

6.1 Dunnett and Sobel's Probability Integrals

129

7 = y1y2 - v,

and 6

= Vyi - 2Pyiy2 + y2 + v(1 - p2).

In the special case yl = Y2 = 0, both (6.3) and (6.4) reduce to the neat expression

= arctan

P(0, 0; v, p)

__p

2

V1

(6.5)

,

_p

which is independent of v and is therefore identical with the corresponding result for the bivariate normal integral. Since the number of terms in (6.3) and (6.4) increases with v, the usefulness of these expressions is confined to small values of v. Dunnett and Sobel (1954) also derived an asymptotic expansion in powers of 1/v, the first few terms of which

yield a good approximation to the probability integral even for moderately small values of v. The method of derivation is essentially the same as that used by Fisher (1925) to approximate the probability integral of the univariate Student's t distribution: Express the difference f (xl, x2; v, p) - f (xl, x2; oo, p) as a power series in 1/v and then integrate this series term by term over the desired region of integration. Setting r

2

=

yi - 2Pyly2 + y2

1-P2

1

Dunnett and Sobel obtained .f (y1, y2; v, P) .f (Y1, y2; 00, P)

r2+ 4

-r) 2

r12

+

(r8

1

5r6

32

1/

7r10

12

13r8

4

8

96 + 24 - r)

( 384 r16

( 6144

r14

17r12

128

144

1

+r) v2 1

v3

77r10 120

1

r) v4 8

+

1 + D(r),

f

say. Thus, the desired probability integral is P(y1, y2; v, P)

=

f (xl, x2; oo, P)dxl dx2 00

CO

/92

/91

J D(r)f(xi,x2;oo,P)dxldx2. (6.6) -00 -00

+J

The first term on the right-hand side of (6.6) is the integral of the bivariate normal pdf, and it has been tabulated by Pearson (1931) with

Probability Integrals

130

a series of correction terms. The second term can be integrated term by term to obtain an asymptotic expansion in powers of 1/v. Dunnett and Sobel gave expressions for the coefficients Ak of the terms 1/v' for k = 1, 2, 3, 4. The first of these coefficients takes the form

Al =

a42O(a)0(y2)

+

b410(b)O

(yi) -

(y2) ID (a)

_yi(y1 4

where 0 and 4) are, respectively, the pdf and the cdf of the standard normal distribution, and

a =

Y1 - Py2

1-p2' y2 - Py1

1-p2 In the special case y1 = y2 = y, (6.6) reduces to V

P(y, y; v, P)

V

= f f f (xi,00x2; oo, P)dxldx2 + `41 +

A2

V

v2

+

A3

+

v3

A4

+

.. .

(6.7)

V4

with the first two coefficients Al and A2 now taking the forms Al

=

YO(y) J (y2

+ 1) (cy) -

y(p'(cy)}

and A2

y4(y) { (3ys

- 7y4 -5 Y2 - 3) (D (cy)

-y-'(cy) [3y4 (c4 + 3c2 + 3) - y2 (c2 + 5) - 3] },

where c =

(1 - p)/(1 + p). In this special case, Dunnett and Sobel

(1954) tabulated numerical values of the coefficients Ak for selected values of p, y, and v. The following table gives the values for p = 0.5

6.2 Gupta and Sobel's Probability Integrals

131

Coefficients of the asymptotic expansion (6.7) for p = 0.5 y

v

Al

A2

A3

A4

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50

4

0.003371 0.008999 0.021983 0.047374 0.079687 0.096254 0.067469 -0.020268 -0.149011 -0.276255

3.00

18

-0.025870 -0.057784 -0.100016 -0.150182 -0.198378 -0.231628 -0.240531 -0.223682 -0.187525 -0.142571 -0.062685

0.003816 0.006868 0.006891 -0.006835 -0.033130 -0.038696 0.052274 0.293449 0.623867 0.858993 0.432592

-0.001050 -0.002155 -0.001879 0.007991 0.036817 0.032808 -0.191482 -0.819219 -1.618705 -1.765249 2.236773

4 6 5

6 6 9

12 13 12

-0.376815

These values can be used to construct tables for the probability integral in (6.7).

6.2 Gupta and Sobel's Probability Integrals Gupta and Sobel (1957) investigated the special case when X follows the central p-variate t distribution with degrees of freedom v and the correlation structure r2j = p = 1/2, i 54 j. If Yl, Y2 i ... , Y,,, Y are independent normal random variables with common mean and common variance v2, and if vS2/o2 is a chi-squared random variable with degrees of freedom v, independent of Y1, Y2,. . ., Y, , Y, then one can rewrite the probability integral as

P(d)

=J

d

/'d

.. . J

f(xl,...,xp;v,p)dxp...dxl

00

= Pr max(Y1iY2, S...,Yp) -Y

< vf2-d1

Pr(Mv; Y
Probability Integrals

132

is derived by fixing Y and S in (6.8) and integrating with respect to MP

P(d) =

f

h(s)

[f: VM 0 y - /ds) dy ds,

(6.9)

where 0 and 4' are, respectively, the pdf and cdf of the standard normal distribution and h is the pdf of the chi-squared distribution with v degrees of freedom. Based on the fact that the pdf 0 admits an expansion

about d = 0, it easy to justify a term-by-term integration of (6.9) to obtain the second expression

P(d)

_

p+1

1

2kdk AkE t(Hk (max(XiX2...X+i))}

CIO

k-o

where

Ak = E

-22

(S)k

r(te)

(6.10)

(z) / r (z)

01

is the kth moment of xv// (provided that k > -v) and Hk is the kth Hermite polynomial defined by C-

d

(- 12)

)k exp

2)

= Hk (x) exp (- 2

2

.

(6.11)

A third expression for P(d) is derived by first expanding 0 about S = o, and then integrating term by term, obtaining

P(d)

=

f

00

DP(y) O (y - v/2-d)

- fdO(') (y - mod) E (S - 1 I

+d20(2)(y-\d)E

f

00

VWO(y -

/

S-1\12

dy

I

fd)dy

-vd(1 - Al)

f

+2d2(1-A,)J

00

(y

-

1 d) 4'P(y)0 (y

- mod) dy

{y2-2/dy+2d2-1}

x 4'P(y)o (y - ,f2-d) dy + .. .

where Al is given by (6.10). Each of the integrals above can be evaluated

by expanding the pdf 0 about d = 0, as was done in (6.9). The fourth

6.2 Gupta and Sobel's Probability Integrals

133

and final expression for P(d) given by Gupta and Sobel (1957) uses the result of Seal (1954) that the distribution of D = (Mp - Y)/a is asymptotically normal as p tends to infinity. It follows directly from Seal's result that the third and higher central moments of D tend to the corresponding moments of the standard normal distribution. Since the coefficients involving v in A_k; in (6.10) tend to unity as v -4 00, it follows that the third and higher central moments of Z = (Mp - Y)/S tend to the corresponding moments of the standard normal distribution as both v and p tend to infinity. It is therefore reasonable to approximate

the distribution of W = (Z - E(Z))/ Var(Z) by a Gram-Charlier expansion in the Edgeworth form, where

E(Z) = A-lap,' and

Var(Z)

= A_2 (ap,2 + 1) - (A_lap,l)2 .

Here, ap,i denotes the ith moment of the largest of p independent standard normal random variables. Using equation (17.7.3) of Cramer (1951) and letting d8 = (vd - E(Z))l Var(Z), Gupta and Sobel obtained

P(d)

= Pr (Z < Vd) = Pr (W < d8) _

-D (d8)

-3

12i (d8) 16x23

0(5)

+ 4i 0(3) (d8) +

- 5i 0(4) (d3) - 35

a4 73

(d8)

(6) (d8) -

2800

018) (d8)

+ .. .

where ak

=

(6.12) K2

is the kth standardized cumulant of Z obtained from the moments around the origin. In a related development, Gupta (1963) studied the above case p = 1/2 and showed that P(d) = P(d; v) satisfies dP (d; v) + v {P(d; v) - P(d; v + 2)} = 0, (6.13) dd which is Hartley's differential-difference equation for the probability in-

tegral of a general class of statistics known as Studentized statistics.

Probability Integrals

134

Using Hartley's solution (obtained using the theory of characteristics), Gupta obtained an approximation for P(d; v) in powers of 1/v and remarked that it can be computed by using the Gauss-Hermite quadrature. Gupta et al. (1985) extended this result for any p > 0 and showed that P(d) satisfies (6.13) in this case too. In this case the approximation for P(d) in powers of 1/v is

P(d)

G(d,... , d) + E Lk (d),

=

(6.14)

k=1

where Lk is the kth correction term and G is the joint cdf of a p-variate normal distribution with zero means, common variance o'2, and the equicorrelation structure rzj = p, i 54 j. Letting G(k) (d) denote the kth-order derivative of G(d, ... , d) with respect to d, the first four correction terms can be written as L1(d)

=

sv2-13a (4)

L2(d)

1 1x(2)

- a(1)

- 10x(3) + 9x(2) - 2x(1) },

s1 {a(6) - 7x(5) + 17x(4) - 17x(3) +6x(2)},

L3(d)

and L4 (d)

=

1

360v4

115c,(')

- 180x(7) + 830x(6) - 1848x(5) + 2015x(4)

-900a(3) + 20a(2) + 48x(1) },

where a (k)

=

1 (k) (d),

k = 1,2,...,8

and the first eight cp(k) (d) are (PO) (d)

= dG(1) (d),

cp(2) (d)

= d2G(2) (d) + dG(1) (d),

= (d) =

cp(3) (d)

d3G(3) (d) + 3d2G(2) (d) + dG(1) (d),

W(4)

d4G(4) (d) + 6d3G(3) (d) + 7d2G(2) (d) + dG(1) (d),

cp(5) (d)

= d5G(5) (d) + 10d4G(4) (d) + 25d3G(3) (d) + 15d2G(2) (d) +dG(1) (d),

cp(6) (d)

=

d6G(6) (d) + 15d5G(5) (d) + 65d4G(4) (d) + 90d3G(3) (d)

6.3 John's Probability Integrals

135

+31d2G(2) (d) + dG(1) (d), (7)

(d)

=

d7G(7) (d) + 21d6G(6) (d) + 140d5G(5) (d) + 350d4G(4) (d)

+301d3G(3) (d) + 63d2G(2) (d) + dG(1) (d), W(8) (d)

= d8G(8) (d) + 28d7G(7) (d) + 266d6G(6) (d) + 1050d5G(5) (d) +1701d4G(4) (d) + 966d3G(3) (d) + 127d2G(2) (d) (6.15)

+dG(1) (d).

Thus the evaluation of P(d) in (6.14) involves that of G(O for k = 0, 1, ... , 8, and we shall discuss in Chapter 8 how the latter can be performed.

6.3 John's Probability Integrals John (1961) provided alternative formulas for the evaluation of the probability integral. Although the method is discussed in detail only for the

bivariate case, it has wider applicability in the sense that it can be adopted to obtain the probability integral of the multivariate t distribution for any dimension.

Let X be a p-variate vector having the central t distribution with degrees of freedom v and correlation matrix R. Using the definition that X can be represented as (Z1 /S, Z2/S,.. . , ZP/S), where Z is a pvariate normal random vector with correlation matrix R and vS2/a2 is an independent chi-squared random variable with degrees of freedom v, one can show that the characteristic function of X is

E (exp (itT X))

= E (E (exp(itTZ/s

IS

= 8))

00 x"/2-1

t(v/2)Jo

exp

(-x -

4xtTR_lt)

dx.

In the case p = 2 with the equicorrelation structure rzj = p, i : j, the above expression reduces to E (exp (it1X1 + it2X2))

1

=

fTO

°°

x"12-1 E

2x) titz

li=o ii! ( x exp { -x - 4x (ti + t2) } dx.

F(v12)

By the inversion theorem, John (1961) derived the corresponding joint pdf as an infinite series of one-dimensional integrals. Integrating the

Probability Integrals

136

infinite series term by term, the probability integral becomes O° pi

1

=

P (yl, y2; v, P)

yv,o (Y1, yz) +

y,.,i (Y1, y2) , E i-1 iI

27r

where 1

yv,o (yi, Y2)

I (v/2) Jo

xV 12-1 exp(-x) 2xy2

x4 (V2XY1

dx

and

r(v1

yv,i (yl, y2)

12)

Jv/2_iexprxf1+i (+ ) J l

xHi_1

(\/Y1Hy2 Z

1

v-v

for i = 1, 2, .... Here, P(.) is the cdf of the standard normal distribution

and Hk denotes the Hermite polynomial of order k defined by (6.11). John provided explicit algebraic expressions for yv,i for i = 1, 2, ... , 6. The first three of them are Yv,1 (yl, y2)

yv,2 (y1,y2)

=

z-v/2 ,

y1y2z-(v/2+1),

and

yv,3 (yl, y2)

=

2 2 2 -(v/2+2) _ 2 z) 1 + v yly2z (yl + y2 z

(v/2+1)

+ z v/z ,

where z = (yi + y2) /v + 1. In principle, explicit expressions for yv,i can

be obtained for any i > 1. To evaluate yv,o, the integration has to be done numerically. John tabulated values of this quantity for v = 11, 12 using Gauss' formula for a numerical quadrature (Kopal, 1955, page 371). He also provided several useful recursion relations. For example, values of yv,o (Y1, y2) for yl negative or y2 negative or both negative can be found from the formulas yv,o (Y1, Y2)

= Tv (y2) - yv,o (-Y1, Y2),

yv,o (Y1, yz)

= T v (y1) - yv,o (y1, -y2) ,

6.4 Amos and Bulgren's Probability Integrals

137

and yv,o (y1, y2)

1 + yv,o (-y1, -y2) - Tv (-yl) - Tv (-y2) ,

=

where Tv is the cdf of the Student's t distribution with v degrees of freedom.

6.4 Amos and Bulgren's Probability Integrals In a widely quoted paper, Amos and Bulgren (1969) derived several representations for (6.2) in terms of series and simple one-dimensional quadratures, together with efficient computational procedures for the special functions used in their numerical evaluation. One of the quadrature formulas given is 1

P =

27r(v + 1)(1 + rye + 72)v/2 xf12

2F1

(1,2V; v+3 2

1

;1-c2cos2(6d9

r((v + 1)/2) V/7rF(v/2)(1 +'y +72)v/2 x 92 I {cos(9 - 0) < 0} cos(9 - 0) v

dB

where 2F1 is the Gauss hypergeometric function, I{} is the indicator function,

F/12 c =

+ 'Y2

1+ry1 +72'

'Y1

= (y2 + yl)

72

=

(y2 - yl)

'

2v' ++ p

01

=

7r - arctan

92

-

it -l- nrrtnn

1-p'

Probability Integrals

138

{

0

arctan (rye/y1), if yl > 0, 7r + arctan (y2/y1) , if yl < 0, A1

=

A2

=

1

1+p'

and 1

1-p

One of the series formulas given is

1rO

P-

(-c)k

r((v + k)/2)

2 fF(v/2) O (1 + yl + y2)"/2 r((1 + k)/2) 02

X

cosk(9 - O)d9.

(6.16)

fol

For the special case v = 1, P can be reduced to the closed-form expression

where 2r sin 0

U

A(1+r2+2rcos =

V

r

1-r2 A(1 +r2 +2rcos0)' Yi + 'Y2

=

1+

1+yl +y2

,

and

A =

P =

1 21r

arctan

tan

(02_lr)

Y1Y2

(V'1+yi +y2

2

7r

+ arctan y1 + arctan y2 +

2

The advantage of these expressions over the ones given by Dunnett and

Sobel (1954) is that these are easier to compute, especially for large

6.5 Steffens' Noncentral Probabilities

139

degrees of freedom. For instance, the integral in 0 in (6.16) can be expressed in terms of incomplete beta functions that are extensively tabulated. Amos and Bulgren (1969) numerically evaluated values of P for all combinations of p = -0.9, -0.5,0,0.5,0.9 and v = 1, 2,5, 10, 25, 50.

6.5 Steffens' Noncentral Probabilities Consider the p-variate noncentral t distribution defined in (5.1). Motivated by the Studentized maximum and minimum modulus tests, Stef-

fens (1970) studied the particular case for p = 2 and R = Ip. In this case, the joint pdf (5.1) reduces to f (X1, X2)

= exp

( ei +e21

X

7rF (v/2)

2

f

°°

1

Cl

k

V LSlxl 1

k!l!V(k+1)/2+1

k=0 l=0 l

(V LS2x2)

°° r((v+k+l)/2+1)

(1 +

2v+ x2v (v+k+l+2)/2 J

where ei = µj/o, are the noncentrality parameters and v denotes the degrees of freedom. The testing procedures involve maximum or minimum values of the components X1 and X2 and the computation of the corresponding probabilities. For this reason, Steffens (1970) derived series representations for probabilities of the form P1 = Pr(I X1 1< A, I X2 1< A) and P2 = Pr(I X1 J> A, I X2 > A). It is seen that P1

= 2exp -

2+2 2

00

)

00

1: E

(e2/2)k (e2/2)l

k!l!B (k + 1/2,1 + 1/2) k=o t=o

7r/4

( sin2k

x fo

v cos21 v+ sin21 v cos2k v)

xIa (k+1+1, 2)dv and P2

=

2(exp

i

+L) 2

00

CO

(f2/2)l EE k!l!B (k + 1/2,1 + 1/2) k=O1=0

fr/4 x

(sin2k v cos21 v+ sin2t v cos2k v) x

{1-Ip (k+1+1, 2)}dv,

where I, denotes the incomplete beta function ratio, a = A2 sec2 v/(v+ A2 sec2 v), and /3 = A2cosec2v/(v + A2cosec2v). Using these represen-

Probability Integrals

140

tations, Steffens estimated values of the critical points A for all combinations of v = 1,2, 5,10, 20, 50, oo and 1i T=2 = 0(1)5 for the significance level 0.05. In a more recent development, Bohrer et al. (1982) developed a flexible algorithm to compute probabilities of the form Pr(cii _ < XP < c21, ... , clp _< Xp < c2p) associated with the noncentral p-variate distribution (5.1).

6.6 Dutt's Probability Integrals Dutt (1975) obtained a Fourier transform representation for the probability integral of a central p-variate t distribution with degrees of freedom v and correlation matrix R

P

(y1,...,yp)

yl f...ff(xi,...,xv;v)dxp...dxi.(6.17)

_

' OPO

Using the definition of multivariate t, one can rewrite (6.17) as 00

2

P (yl,...,yp)

J

zv-1 exp (-z2/2) G (yl, ... , yp) dz, (6.18)

where yk = ykz/-\,fv-, k = 1,... , p and G is the joint cdf of the multivari-

ate normal distribution with zero means and correlation matrix R. In the case yk = 0, one has P independent of v and

P(y1,...,yp)

= G(0,...,0).

Explicit forms of G for p = 2, 3, 4 in terms of the D-functions are given in

Dutt (1973). The D-functions are integral forms over (-oo, oo) defined by 00

Dk (t1, ... , tp; R)

(27r)k

dk

f.... foo Si...Sk

(itlsl _s k

xexp

l-0

2

dsk

dsl,

l=0

where the first five dk are d1

=

1,

d2

=

d12,

d3

= =

d12+13+23 - (d12 + d13 + d23)

d4

-d12+13+23+14+24+34 + d12+13+23 + d12+14+24 + d13+14+34

6.6 Dutt's Probability Integrals

141

+d23+24+34 - (d12 + d13 + d23 + d14 + d24 + d34) d5

=

-dl2+13+23+24+34+15+25+35+45 + d12+13+23+14+24+34 +d12+13+23+15+25+35 + d12+14+24+15+25+45 +d13+14+34+15+35+45 + d23+24+34+25+35+45

- (d12+13+23 + d12+14+24 + d12+15+25 + d13+14+34 +d13+15+35 + d14+15+45 + d23+24+34 + d23+25+35

+d24+25+45 + d34+35+45) + d12 + d13 + ... + d45,

and dpi 9i +...+pm 9m

=

1 - exp {- (rpi 9i spi S9,

+ ... + rpm qm Spm S9m) }

Using the notation Dk: jl ,...,jk

Dk {tj1,...,tjk; R (tjl,...,tjk)},

where R (tj1, ... , tjk) is the correlation matrix based on the subscripts jl, ... , jk, Dutt (1973) provided the following explicit forms for G G (tl, t2) G (t1i t2, t3)

=

=

{1 - 4' (tl)} {1 -'t (t2)} + D2:1,2,

{1 - (D (t1)} {1 -"D (t2)} {1 -'D (t3)} + {1 - 1 (t1)} D2:2,3 + {1 - 1 (t2)} D2:1,3 + {1- 4 (t3)} D2:1,2 + D3:1,2,3,

and 4

G (t1, t2, t3, t4)

_

11 {1 -

(tk)} + 11 - 4' (tl)} {1 -'P (t2)1 D2:3,4

k=1 +{1-'I(tl)}{1-4 (t3)1 D2:2,4

+{1-4) (t2)}{1-'D (t3)} D2:1,4

+{1-c1(tl)}{1-$(t4)}D2:2,3 + {1 - 4) (t2)} 11-4) (t4)} D2:1,3 + {1 - 4) (t3)} {1 - (D (t4)} D2:1,2 + {1 - 4) (t1)} D3:2,3,4 {1 - 4) (t2)} D3:1,3,4

+ {1 - 4) (t3)} D3:1,2,4 + {1 - 4) (t4)} D3:1,2,3 +D4:1,2,3,4

A much simplified representation for G in terms of the error function, erf(.), and integral forms over (0, oo), denoted as the D* functions, is

Probability Integrals

142

given in a later paper by Dutt (1975). These D*-functions are defined by

Dk tl,...,tP;R

[00

2

(27r)k

... [00 -d*k k

(_s?/2) dsk " ' dsl

x exp

,

(6.19)

1=0

where for the first few k are di

= sin(tisi),

d2

= e_12 cos1_2 -e12 cos1+2,

d3

= el2+13+23+14+24+34 COS1+2+3+4 +e12-13-23-14-24+34 COS_1_2+3+4

+e-12+13-23-14+24-34 COS_1+2-3+4 +e-12-13+23+14-24-34 COSl_2_3+4 -e-12-13+23-14+24+34COS_1+2+3+4 -e-12+13-23+14-24+34 COS1_2+3+4 -e12-13-23+14+24-34 COS1+2+3+4

-el2+13+23-14-24-34 COS1+2+3-4

and for notation ePl gl+...+p-q-

exp { - (rp1 q1 sp1 sq1 + ... + rpm q,n spm sq,,,) J ,

= COSP1+ +P-

sin (tP1 sP1 + ... + tpm Sp_) ,

= COS (tn, sp, + ... + tPm sp-)

(A negative sign on the index pl ql corresponds to +rp1 q, sp1 sq1 and -pl corresponds to -ti, sp, .) Important special cases of these functions are Di (y)

=

1 erf 2

D2 (0, 0; R) =

y V'(2

1 arcsin (r12) ,

and D* (0; R)

0,

=-

for k odd.

Using the abbreviation that Dk:jl,...,jk

-

Dk 1.

21'

S ..

t3k ; R (tjl, ... , tjk )}

,

6.7 Amos' Probability Integral

143

Pp-22 P

Dutt (1975) provided the following representation for G

(1) mp

G(t1,...,tP)

Di:k+

k=1

(j)p-3

\

D2:kl

k
P

2k

< <m= 1

Hence, by (6.18), the computation of P in (6.17) can be achieved by successive applications of the Gauss-Hermite quadrature formula using only positive Hermite zeros (Abramowitz and Stegun, 1964, page 924). There are several advantages for this approach. First, it is not necessary to invert the correlation matrix. In addition, (6.19) permits the use of Gauss quadrature formula that are remarkably effective in estimating the value of an integral from a few points, provided that the integral excluding the weighting function can be accurately approximated by a polynomial. Moreover, often the integrand separates as a product of two functions, one depending only on correlation coefficients and the other on the original limits of integration. For selected correlation structures and several values of v and y = yk,

k = 1,... , p, Dutt (1975) computed values of P accurate up to six decimal places.

6.7 Amos' Probability Integral For the equicorrelation structure rid = p, i A j considered by Gupta and Sobel (1957) and Gupta (1963) - but with the common p taken to be any positive real number less than 1 - Amos (1978) derived the following simpler expression for the probability integral P(d)

=

1°°

1)/2)

2v-312r ((v

2 7r (1+b )v/2

J

exp

oo

d22 f1

x,V(x)erfc clx(-) dx, where

is the complementary error function defined by erfc(x)

=

00 fexp(_z2)dz

and a, b, c, d are constants given by

a =

-p

ri P

,

(6.20)

Probability Integrals

144

d

b

pv' ab

d =

1+b2*

The reduction to (6.20) was obtained by means of a relationship between the parabolic cylinder function and the complementary error function.

Amos (1978) suggested computing the integral (6.20) by locating the x0 for which the derivative of the integrand is zero and then summing quadratures on intervals of length h to the left and right of x0 until a limit of integration is reached or the truncation error is small enough. The motivation for this procedure comes from the fact that x0 can vary widely with extreme parameter values, and h, which estimates the spread

of the integrand, can be small or large. Thus, x0 and h accommodate the parameters, producing meaningful results by preventing quadratures over tails that are negligible or preventing gross misjudgments of the scale of integration. Letting g(x) denote the integrand of (6.20), Amos (1978) showed that the derivative of logg(x) decreases monotonically from no to -oo as x traverses (-oo, oo), guaranteeing a unique root x0 of g'(x) = 0.

6.8 Fujikoshi's Probability Integrals Fujikoshi (1988) provided asymptotic expansions as well as error bounds

for the probability integral (6.17) when the correlation matrix R = I, , the p x p identity matrix. Specifically, letting ab,j (y1, ... ' yn)

=

dsj

{,4 (s-1i2y1) ...4, (s-1,2yr)}

where 6 = -1, 1, and (D denotes the cdf of the standard normal distribution, Fujikoshi established the following approximation for the probability integral k-1 1

P

(y1,...,yr)

_

(y1)...

(yr) +

j=1

xE

/ I

7

2

X2

I

\\v///

-1

ab>j (y1,...,yr)

jl (6.21)

6.9 Probabilities of Cone

145

which we shall denote by As,k (yl, ... , yp). Fujikoshi also derived uniform and nonuniform error bounds for this approximation. Under the assumptions that sup Ias,k (yi, .. , , yp)I < 00,

ab,k

Y

and

(- /

2

E (v

E

<00,

k

the uniform bound takes the form sup I P (yi, ... , yp) - As,k (yi, ... Y

klaakE C

yp) I

(v)Under \v)V ,

(

the assumptions that aa,k(l)

= sup (1+ II y II`) Ias,k (yi, ... , yp)l < 00 Y

and

E

\ k+1/2

(vV

< 00,

k

(-

E

< 00,

/

the nonuniform bound takes the form P (yl, ... , yp) - Aa,k (yl, ... , yp) (1+ II Y

aa,k(l)E

(v2\ l/ J

2

2

v-1

k

+ xv2

k .

Clearly the latter bounds are improvements on the uniform bounds in the tail part of the multivariate t distribution. In the case p = 1, these results provide useful approximations for the univariate Student's t distribution - see Fujikoshi (1987) and Fujikoshi and Shimizu (1989). The special case of (6.21) for yj = y has been investigated more recently by Fujikoshi (1988, 1989, 1993), Fujikoshi and Shimizu (1990), and Shimizu and Fujikoshi (1997).

6.9 Probabilities of Cone Consider the p-dimensional set Ar(c)

= {x : zTx < r 11 z II, all z in E(c) } ,

(6.22)

146

Probability Integrals

Fig. 6.1. The sets A,.(c) and E(c) fl {II z 11= r} in two dimensions

where E(c) = {z : zl > c 11 z 11}, 11 z 11= v7 z, and c is a nonnegative

constant. The set E(c) is the cone, with vertex at the origin, which intersects origin-centered spheres in spherical caps. This is illustrated in Figure 6.1 for p = 2. Bohrer (1973) studied the analytical shape of A, (c) and the associated probability p(c, r, p, v)

= Pr (X E A,.(c))

when X has the p-variate t distribution with mean vector 0, covariance matrix o,2Ip, and degrees of freedom v. The evaluation of p(c, r, p, v) is of statistical interest and use in the construction of confidence bounds (Wynn and Bloomfield, 1971, Section 3; Bohrer and Francis, 1972, equation (2.3)) and in testing multivariate hypotheses (Kudo, 1963, Theorem 3.1, Section 5; Barlow et al., 1972, pages 136ff, 177).

6.9 Probabilities of Cone

147

As regards the shape, Bohrer showed that every two-dimensional section of Ar containing the zl-axis is exactly the two-dimensional version of Ar illustrated in Figure 6.1. Thus, Ar is the solid of revolution about

the zl-axis that is swept out by the Ar in Figure 6.1. To express this more precisely in mathematical terms - for an p x 1 vector v - define polar coordinates R and µv = {Bvi}, with -7r < Ovi < 7r, by

= Rvcos0 i,

v1

i-1

Rv cos Bvi [J sin 9Vj,

Vi

j=1 i = 2,...,p - 1,

and i-1

VP

= RV 1 sin9vj. j-1

Also define = arccosc,

0*

T1

= {x : Bt1 - 0* E (0, 7r/2], Rt cos (0t1 - 9*) <

T2

T3

= {x: 10t1I <0*,Rt
=

{x : Bt1 + 0* E [-ir/2, 0), Rt cos (0t1 + 0*) < r} ,

and T4

=

{x : I0t1 I > 0* + 7r/2} .

Then the set Ar is the union of the disjoint sets T1i ... , T4. As regards evaluating the probability p(c, r, p, v), Bohrer (1973) derived the following expression p(c) r, p, v)

=

k(02) Pr (Fr < Q2 k 2k(7/2)

)

+

:B (i+

k

2 1

(k (7r/2) *)

p-2- 11

(p 2)

7

2

xcj (1

- c2) (r-2-j)/2 Pr (F_2_,v < r2

Probability Integrals

148

where k(9) is given by (2mm!)2

k(9)

=

(1 - COs 9) -

COS 9 sinzm 9

p!

P M_ 1

-uE

sin2(m-t) 9

p-21

p - 2j

cos 8

11p+1-2j

when p = 2m + 1 is odd and by k(9)

_

sine-1-2m

(p -1)!9 2P-1(m - 1)!m!

-E

0 COS 9

p sine-1-z1 0c so n

p-21

1==11

p+1-2j j11p+2-2j

when p = 2m is even. The statistical questions that motivate this work ask what radius r is required so that p(c, r, p, v) = a for preassigned values of a. For p < 5, Bohrer (1973) provided tables of these percentiles for a = 0.95 and 0.99 and for a range of (c, v) pairs.

6.10 Probabilities of Convex Polyhedra It is well known (Nicholson, 1943; Cadwell, 1951; Owen, 1956) that probabilities of polygons under bivariate normal distributions can be evaluated in terms of probabilities of right-angled triangles with vertices (0, 0), (Y1, 0), (yl, y2), yj > 0, j = 1, 2 under bivariate normal distributions with zero correlation. John (1964) proved an analogous result that probabilities of polygonal and angular regions for a given bivariate t distribution can be expressed in terms of V, (Y1, Y2), the integral of f (xl, xz; v)

=

I ((v + 2)/2) xl + vrrl,(v/2) 1+

x2 }_(V+2)/2

over the right-angled triangles with vertices (0, 0), (yr, 0), and (yi, y2) John (1964) also provided several formulas for evaluating Vv(y1iy2). A formula in terms of the incomplete beta function is Vv (y1, Y2)

(Y2) 1 arctan \ yi ylcv/2 47r

v-+y 1 k=O

Zck Bu+k+22(6.23) (2

6.10 Probabilities of Convex Polyhedra

149

where V c

--

Y21

v+yi v+y1 +y2 2

and

=

B,, (a, b)

fwa_1(1_w)bldw

is the incomplete beta function. This series converges slowly unless yl is large in relation to v. In the two cases v odd and v even, (6.23) can be reduced considerably. If v = 2m for a positive integer m, then m-1

=

V2m (y1, y2)

1:

47

1

k+

1

(6.24)

2' 2

k=O

while if v = 2m + 1 for a nonnegative integer m, then

2

=

V2m+1 (y1, y2)

7r

arctan I

+

(1 ,

Y2

yl

c(1-c)

47r

M-:

B"

1)

2 2

ck B, (k+1, 211/

I

4ir

(6.25)

k=O

where V

=

V (v+y1 +y2) V (v + yi + y2) + y1 2

An attractive feature of (6.24) and (6.25) is that, when utilizing them for evaluating V2m and V2,,,+i, they are already evaluated for lower values of m also. If one performs the summations in the order indicated in the formulas, the addition of each term will yield values of V2,,,, or V2,+, for the next higher value of in. This feature makes it particularly suitable for use in preparing tables. A second formula for V, (Y1, y2) given in John (1964) is an expansion in powers of 1/v V (y1, y2)

= V. (y1, y2) -

exp (-yi/2) °° v-k 27r

k=1

k!

Uk (yl, y2) (6.26)

Probability Integrals

150

where the first three Uk are given by 4

U1 (y1, y2)

= 4 W1 (y1, Y2), 2

U2 (yi, y2)

=

- 3W2 (yl, y2) +

Y1'

16

Ws (yl, y2)

4

2

y$

4W3 (yl, y2) - 4 W4 (yl, yz) + 64 W5 (y1, y2)

y2/y1

Wv (yi, y2)

=

/

2t2 \

(1 + t2) ' exp l - y 2 I dt.

1

The term V... in (6.26) is the integral of exp { - (y1 + y2) 1/(27r) over the right-angled triangle with vertices (0, 0), (yl, 0), and (0, y2). The method of derivation for (6.26) is similar to the classical method employed by Fisher (1925) for expanding the probability integral of Student's t. Despite the complexity of (6.26) over (6.23), (6.26) should be preferred if v is sufficiently large. The first two or three terms of (6.26) then can be expected to provide fairly accurate values of V. John (1964) also provided a recurrence relation and an approximation for Vv(y1i Y2); the latter proved to be satisfactory only when either v is

too small or y2/yl is too large. In a subsequent paper, John (1966) extended this result to higher dimensions, by showing that the probabilities of the p-dimensional convex polyhedra with vertices (0, 0, 0, 0, ...,

0), (y1, 0, 0, 0, ..., 0), (y1, y2, 0, 0, ..., 0), ..., (yi, y2, y3, y4, ..., Yp), hj > 0, j = 1, 2, ... , p under a p-variate t distribution with v degrees of freedom can be expressed in terms of the function V, (Y1, Y2,

yp), the

integral of the p-variate t pdf f (x1, X2.... ) xp; v) 1+x1+x2+...+xP

F((v+p)/2) (v7r)pl2F(v/2)

-(v+p)l2

v

over the same p-dimensional convex polyhedra. John also provided an important asymptotic expansion in powers of 1/v connecting Vv(yl, y2i . .., yp) with V (y1i y2i ... , yp), the integral of the p-variate normal pdf f (xi, X2.... , xp; oo)

_

(27r)-p/2

exp {- (xi + x2 + ... + xP) /2}

6.10 Probabilities of Convex Polyhedra

151

over the same polyhedra discussed above. Up to the order of the term O(1/v2), the expansion is V (y1, y2, ...

,

Yp)

V (y1, y2, ... , Yp) +

4v

1y1y2f (y1, y2) V (y3, y4, ... , Yp)

-yl (1 + yi) f (y1) V (y2, y3, ... , yp) } +96v2 {3y1y2y3y4f (y1, y2, y3, y4) V W5, ys, ... , yp) -Y1Y2Y3 (2 + 9yi +6 Y2 + 3y3) f (y1, y2, y3) V (y4, ... , yp)

-Y1Y2 (3 + 5yi + y2 - 9yi - 9y1 2 - 3y2) XV (y3,...,yp) f (Y1, Y2)

+yl (3 + 5yi +7 Y4

- 3y6) f (YO V (y2, ...

Yp)

}

+olv/. In this formula, V (y,,, y,,.,,+1, ... , yp) is to be replaced by 0 if m > p + 2

and by 1 if m = p + 1. In principle, there is no difficulty in determining further terms of this expansion, but the coefficients of higher powers of (1/v) have rather complicated expressions. Other useful results given by John (1966) include recursion formulas connecting V (yl, y2i ... , Yp) with Vv±2 (y1, y2, ... , Yp)

More recently, several authors have looked into the problem of computing multivariate t probabilities of the form

P = f f (x; v) dx,

(6.27)

A

where X has the central multivariate t distribution with correlation matrix R and A is any convex region. Somerville (1993a, 1993b, 1993c, 1994) developed the first known procedures for evaluating P in (6.27). Let MMT be the Cholesky decomposition of R (where M is a lower

triangular matrix) and set X = MW. Then W is multivariate t with correlation matrix Ip. If one further sets r2 = WTW, then F = r2/p has the well known F distribution with degrees of freedom p and v. Let A be the region bounded by p hyperplanes and described by

GW < d, where G = ( 9 1 , . , gp) and the jth hyperplane is g W = d1. For a random direction c, let r be the distance from the origin to the boundary

Probability Integrals

152

of A, that is, the smallest positive distance from the origin to the jth plane, j = 1,. .. , p. Then an unbiased estimate of the integral P in (6.27) is

Pr (F < r2/p)

(6.28)

.

To implement the procedure, Somerville chose successive random directions c and obtained corresponding estimates of (6.28). The value of P was then taken as the arithmetic mean of the individual estimates. Somerville (1997, 1998b) provided the following modification of the

above procedure. Let r* be the minimum distance from the origin to the boundary of A, that is, the smallest of the r for all random directions c. Divide A into two regions, the portion inside the hypersphere of radius

r* and centered at the origin, and the region outside. The probability content of the hypersphere is Pl

= Pr (F < r*2/p)

,

and this can be estimated as in Somerville (1993a, 1993b, 1993c, 1994).

If E(v) and e(v), respectively, denote the cdf and the pdf of v = 1/r (the reciprocal distance from the origin from and to the boundary of A), then the probability content of the outer region is 1/r*

P2

=

E( v)e(v)dv. fo

Since F = r2/p, the pdf of v is e(v)

=

vv-1

2vvl2r ((v + p)/2)

r (v/2) r (p/2)

(1 +

vv2)(v+r)/2

The strategy is to use some numerical method to estimate E(v) and then evaluate the integral P2 using the Gauss-Legendre quadrature. The approaches of Somerville (1997, 1998b) differ in that Somerville (1997) applied Monte Carlo techniques to estimate E(v) while Somerville (1998b) used a binning procedure. It should be noted, however, that an approach similar to these had been introduced earlier by Deak (1990). Somerville (1999a) provided an extension of the above methodologies

to evaluate P in (6.27) when A is an ellipsoidal region. This has potential applications in the field of reliability (in particular relating to the computation of the tolerance factor for multivariate normal populations) and to the calculation of probabilities for linear combinations of central and noncentral chi-squared and F. In the coordinate system of the transformed variables W, assume, without loss of generality, that

6.10 Probabilities of Convex Polyhedra

153

the axes of the ellipsoid are parallel to the coordinate axes and the ellip-

soid has the equation (w - u)TB-1(w - u) = 1, where B is a diagonal matrix with the ith element given by bi. If the ellipsoid contains the origin, then for each random direction c there is a unique distance r to the boundary. An unbiased estimate of P is then given by

Pr (F < r2/p) . If the ellipsoid does not contain the origin, then, for a random direction, a line from the origin in that direction either intersects the boundary of the ellipsoid at two points (say r > r*) or does not intersect it at all. If the line intersects the boundary, then an unbiased estimate of P is given by the difference

Pr (F < r2/p) - Pr (F < r2/p). If the line does not intersect the ellipsoid, an unbiased estimate is 0. As in the first procedure described above, this is repeated for successive random directions c, each providing an unbiased estimate. The value

of P is then taken as the arithmetic average. A modification of this procedure along the lines of Somerville (1997, 1998b) is described in Somerville (1999a). Somerville (1999b) provided an application of his methods for multiple

testing and comparisons by taking A in (6.27) to be

A = {xERP:maxcTx
cEB,

where B is the set of contrasts corresponding to the different hypotheses

and q > 0. The purpose is to calculate the value of q for arbitrary R and v and arbitrary sets B such that the probability content of A has a preassigned value ry. Somerville and Bretz (2001) have written two Fortran 90 programs (QBATCH4.FOR and QINTER4.FOR) and two SAS-IML programs (QBATCH4.SAS and QINTER4.SAS) for this purpose. QINTER4.FOR and QINTER4.SAS are interactive programs,

while the other two are batch programs. A compiled version of the Fortran 90 programs that should run on any PC with Windows 95 or later can be found at http://pegasus.cc.ucf.edu/-somervil/home.html

These programs implement the methodology described above to evaluate the probability content of A (A Fortran 90 programs MVI3.FOR used to evaluate multivariate t integrals over any convex region is described in

Probability Integrals

154

Somerville (1998a). An extended Fortran 90 programs MVELPS.FOR to evaluate multivariate t integrals over any ellipsoidal regions is described in Somerville (2001). The average running times for the latter program

range from 0.075 and 0.109 second for p = 2 and 3, respectively, to 0.379 and 0.843 second for p = 10 and 20, respectively.). The so-called "Brent's method," an interactive procedure described in Press (1986), is used to solve for the value of q. The time to estimate the q values (with a standard error of 0.01) using QINTER4 or QBATCH4 range from 10 seconds for Dunnett's multiple comparisons procedure to 52 seconds for Tukey's procedure, using a 486-33 processor. A problem that frequently arises in statistical analysis is to compute (6.27) when A is a rectangular region, that is,

P-

61

Zb2

by

f (x1i x2, ... , xp) dxp . . . dx2dxl. (6.29)

. . . al

ap

2

Wang and Kennedy (1997) employed numerical interval analysis to com-

pute P. The method is similar to the approaches of Corliss and Rall (1987) for univariate normal probabilities and Wang and Kennedy (1990)

for bivariate normal probabilities. The basic idea is to apply the multivariate Taylor expansion to the joint pdf f. Letting e3 = (aj + bj)/2, the Taylor expansion of f at the mid point (c1, C 2 ,- .. , cp) is

f (x1,x2,...,xp)

_

p

m-1 k=0

akf (Cl, C2, ... , cp) (x kl ! ... kp! axkl axkz ... axkp 1 2 p i=1 1

]k[

1

8mf (S t1, S2t,

]m[

t)

. ., Sp

E ml l ... mpl ax1 1 8x22 ... .

axmp r

- c)

kj

p

11 (x - cj )m' 7-1 , (6.30)

where L;j is contained in the integration region [aj, bj] and ]k[ denotes

all possible partitions of k into p parts. For example, in the case p = 3, ]2[ will result in 6 possible partitions of `2' into {k1, k2, k3}: {0, 0, 2}, {0,1,1}, {0, 2, 0}, {1, 0, 1}, 11, 1, 0}, and {2, 0, 0}. The main problem with computing (6.30) is the presence of high-order partial derivatives of f. Defining (f)k,k2...kp

=

ak,+k2+ +kp f

1

k1!k2!

k p!

kp k2 axk1, axe ...axp

(6.31)

6.10 Probabilities of Convex Polyhedra

155

Wang and Kennedy derived the following recursive formula

xTR-lx

(f

)k1k2...kP

-1

I \1 + k1

X 1

kP

k2

`... p2v(kl-ll)+11

1=01 2= 0

1

(f)1112...1P

P= 0

xTR-1 x1 V

J PC1 -tl,'2-t2....-Xp-1P

With regard to the last quadratic term, it should be noted that higher than second-order partial derivatives are all zero. T o carry out the computation of (6.31) for a given (k1i k2, ... , kr), one can

first let one lj be k1 - 1 (if this kj 54 1) and all the other l1's be their corresponding kg's;

next let l, and is be k,. - 1 and k9 - 1, respectively (if k,. # 1 and k, # 1), while all the other lj's take their corresponding k1's;

finally, let some lj be kj - 2 (if kj > 2) and all other la's be the corresponding kg's.

The total number of terms that contribute to computing (f )k1 k2...kp is at

most p(p+3)/2. Compared to the multivariate normal distribution, this number is larger (Wang and Kennedy, 1990). The following table gives the running times and the accuracy for computing (6.29) with v = 10.

Running time and accuracy for computing P in (6.29) p

10 9

8 7

6 5

Running time (min)

aj = -0.5 aj = -0.4 bj = 0.5

bj = 0.4

80 70 85 90 110 180

3 sig 10 sig

0 sig 3 sig 8 sig

aj = -0.3 aj = -0.2 bj = 0.3

bj = 0.2

2 sig 3 sig 5 sig 8 sig

4 sig 7 sig 10 sig

Another point to note about Wang and Kennedy's method is that when the integration region is near the origin it works better for larger v, while

Probability Integrals

156

when the integration region is off the origin it works better for smaller v.

The main problem with Wang and Kennedy's (1997) method is that the calculation times required are too large even for low accuracy results (see the table above). Genz and Bretz (1999) proposed a new method for computing (6.29) by transforming the p-variate integrand into a product of univariate integrands. The method is similar to the one used by Genz (1992) for the multivariate normal integral. Letting MMT be the Cholesky decomposition of R, define the following transformations P

Xi

Mj,kyk, k=1

Yj

=

Ui

Ui

I-1 y2k V + Ek=1

v+j-1

=

and

= dj + Wj (ej - dj) ,

Z3

where TT denotes the cdf of the univariate Student's t distribution with degrees of freedom T,

a

dj

=

(aaj),

ej

=

(bj)

=a

v+j -1 3

b=b

_1 y2'

V + ,k=1yk

v+j-1 1

V + Ek=1 yk2 1

a _

,

a.7 - Ek=1 Tnj,kYk mj,j

6.10 Probabilities of Convex Polyhedra

157

and

Applying the above transformations successively, Genz and Bretz reduced (6.29) to

P=

1

1

1

(el _ dl) I (e2 - d2) ... I (ep - dp) 0 1

J0

dw

(6.32)

f1

i

J

J

f (w) dw.

(6.33)

The transformation has the effect of flattening the surface of the original

function, and P becomes an integral of f (w) = (el - d1) ... (ep - dp) over the (p - 1)-dimensional unit hypercube. Hence, one has improved numerical tractability and (6.33) can be evaluated with different multidimensional numerical computation methods. Genz and Bretz considered three numerical algorithms for this: an acceptance-rejection sampling algorithm, a crude Monte Carlo algorithm, and a lattice rule algorithm. Acceptance-rejection sampling algorithm: Generate p-dimensional uniform random vectors w1i w2, ... , wN and estimate P by N

1

P = nr

h(My,), 1=1

where

h (x)

=

1

if aj < xj < bj, j = 1, 2.... , p,

0

otherwise

and

v + k-=1 yk 1

)

Tv+,7-1 (w1,j

L+j-1

2

,

j = 1,2,...,p, l = 1,2,...,N. A crude Monte Carlo algorithm: Generate (p-1)-dimensional uniform random vectors w1, w2i ... , wN and estimate P by

I' =

1

N f (WI)

AT l=1

an unbiased estimator of the integral (6.33).

Probability Integrals

158

A lattice rule algorithm (Joe, 1990; Sloan and Joe, 1994): Generate (p - 1)-dimensional uniform random vectors w1i w2, ... , wN and estimate P by 1

2(?z+wi1-1p lq

Here N is the simulation size, usually very small, q corresponds to the

fineness of the lattice, and z E RP-1 denotes a strategically chosen lattice vector. Braces around vectors indicate that each component has to be replaced by its fractional part. One possible choice of z follows the good lattice points; see, for example, Sloan and Joe (1994).

For all three algorithms - to control the simulated error - one may use the usual error estimate of the means. Perhaps the most intuitive one of the three is the acceptance-rejection method. However, Deak (1990) showed that, among various methods, it is the one with the worst efficiency. Genz and Bretz (2001) proposed the use of the lattice rule algorithm. Bretz et al. (2001) provided an application of this algorithm for multiple comparison procedures. The method of Genz and Bretz (1999) described above also includes an efficient evaluation of probabilities of the form b

P = J g(x)f(x)dx, a

where g(x) is some nuisance function. Fortran and SAS-IML codes to implement the method for p < 100 are available from the Web sites with URLs http://www.bioinf.uni-hannover.de/-betz/ and

and http://www.sci.wsu.edu/math/faculty/genz/homepage.

6.11 Probabilities of Linear Inequalities Let X be a random variable characterizing the "load," and let Y be a random variable determining the "strength" of a component. Then the probability that a system is "trouble-free" is Pr(Y > X). In a more complicated situation, the operation of the system may depend on a

6.11 Probabilities of Linear Inequalities

159

linear combination of random vectors, say ai X1 + a2 X2 + b, and the probability of a trouble-free operation will be

Pr(a1Xi+a2X2+b> 0),

(6.34)

where Xj are independent k3-dimensional random vectors, aj are kjdimensional constant vectors, and b is a scalar constant. Absusev and Kolegova (2001) studied the problem of constructing unbiased, maximum likelihood, and Bayesian estimators of the probability (6.34) when Xj is assumed to have the multivariate t distribution with mean vector µj and correlation matrix Rj. If X1 1, ... , X1n, and X 2 1 ,-- . , X2n2 are iid samples from the two multivariate t distributions, then - in the where case both µj and Rj are unknown - it was established that the unbiased and the maximum likelihood estimators are T Or (a1 X1 + a2T X2 + b > 0)

_

r (n1/2) r (n2/2) 7rr ((n1 - 1)/2) r ((n2 - 1)/2) 2

x f 1j=1 (1 - v )(n3)/2 dvldv2 2

and ai Xn, + a2 Xn2 + b

Pr(a1X1+a2X2+b>0)

al Snl+l al + a2Sn2+1a2

respectively, where c1

=

vj? <1,j=1,2, 2

2

1: vj njaj Sni+laj + E a,Txj + b > 0 j=1

j=1 ni 1

= _E ni Xm' m=1

ni+1

(nj + 1) xj

=

1: xjm, M=1

ni+1

(nj + 1) Sni+l

(Xjm - Xj) (Xj.m -;)T ,

= m=1

Probability Integrals

160

and x,,,+i = x. A Bayesian estimator of (6.34) with unknown parameters µj and Rj and the Lebesgue measure p(O)dO = dµdR was calculated to be

F ((nj - kj)/2)

2

PrB (aTX1 + a2 X2 + b > 0)

11 j_1 7rF ((nj - 1)/2) (nj + 1)k' 2

X

F

((nj

-k -

nJ

2

x

1 )/ 2 )

j

11 F ((nj - 2kj - 1)/2) j=1 g

[J (1 -

J 2 j=1

Z3?)

2

dzldz2i

where 2

Q2

=

z? < 1, j = 1, 2, E zj

2

nja

j=1

R. + b > 0 j=1

This Bayesian estimator is biased and is related to the unbiased estimator via the relation PrB (aTX1 + a2 X2 + b > 0)

= APr (aTX1 + a2 X2 + b > 0) ,

where

A = 71 F ((nj - kj - 1)/2) F ((nj - k)/2) (nj + 1)k j=1

F (nj - 2kj - 1)/2) F (nj/2)

nk)-

The coefficient A can be expanded as k A = 1+n-nkk+Ol I) n2

I

where n = max(nl, n2) and k = max(kl, k2). `Therefore, the Bayesian estimator is asymptotically unbiased as n -> oo. Substantial literature is now available on problems concerning probabilities of the form (6.34) for various distributions. For a comprehensive and up-to-date summary, the reader is referred to Kotz et al. (2003).

6.12 Maximum Probability Content Let X be a bivariate random vector with the joint pdf of the form

f (x)

= g ((x - µ)T R-1 (x - µ))

,

(6.35)

6.13 Monte Carlo Evaluation

161

which, of course, includes the bivariate t pdf. Consider the class of rectangles

=

R(a)

{(x1,x2) : lxii < a, 1x21 < A/(4a)}

with the area equal to A. Kunte and Rattihalli (1984) studied the problem of characterizing the region R in this class for which the probability P(R(a)) = Pr(X E R(a)) is maximum. As noted in Rattihalli (1981), the characterizations of such regions is useful for obtaining Bayes regional estimators when (i) the decision space is the class of rectangular regions and (ii) the loss function is a linear combination of the area of the region and the indicator of the noncoverage of the region. It was shown that, for any fixed ) > 0, the maximal set is

{(x1ix2):Ixl-All
A F 4

Here, ri3 denotes the (i,j)th element of the inverse of R. In particular, if p = 0, r12 = r21 = p and I p j< 1 in (6.35), then P(R(a)) is increasing for a < VA-/2 and is decreasing for a > vfA-/2.

6.13 Monte Carlo Evaluation Let X be a central p-variate t random vector with correlation matrix R and degrees of freedom v. Vijverberg (1996) developed a family of simulators of the multivariate t probability p = Pr(X < Xo) based on Monte Carlo simulation and recursive importance sampling. We shall provide the basic steps of this rather complicated but powerful procedure.

Define Z = AX, where A is an upper triangular matrix such that ATA = R. Then it is well known that the pdf of Z can be expressed as a product of univariate Student's t pdfs f (z)

vk

vk

=

=

ftfi(zk;,vk),

= v + p - k,

k = 1,2,...,p,

y2

v +

+1 + ... + yp vk

k = 1 2,... p - 1,

Probability Integrals

162

2

Qn

=

1,

and

fl (x; 02, v)

F ((v + 1)/2) 1 x2(v+1)/2 11 + v / o, r' (v/2) Q2

=

We shall denote by F, (x; Q2, v) the cdf corresponding to fl. For conve-

nience denote A-1 = B = bij, where B is an upper triangular matrix with bpp = 1 and bjj > 0 for all j. Then, since the integral over X covers the region X < Xo, the integral over Z is determined by the inequality BZ < Xo, and the bounds can be written as zp

<

xPo

= zn, and P

Zk

<

1

b-k (XkO

-

bkizi i=k+1

zko (xko, zk+,, ... , ZP)

for k = 1, 2, ... , p - 1. Utilizing this transformation, the probability p can be written as p = Jp, where ykO

Jk

J-

fl (zk;

2

(6.36)

o'k, Vk) Jk-ldzk

00

f

ZkO

F, (zko; a k , Vk) Jk-1fi (zk; 0.2, Vk) dzk 00

2 Efi F1 (zko; 0.2, Vk) Jk-1

,

k = 2,3,... , p,

where c

f, (zk;

2 k,

Vk)

fl

_

zk;

2 0.2,

Vk)

/ F1 (zk0; 0"k, Vk)

is the univariate unconditional t pdf for zk < zko and J1 = F1(zio; 0'1, v1). Hence, J k is the probability over the range of ( Z 1 ,-- . , zk) conditional on the values for (zk+1, ... , zp).

The Monte Carlo simulation starts off by drawing random values of zP from the distribution zpo, up, vp), which we shall denote by zp,r, , R. r = 1,.. . Each of these yields a different bound zp-1,o,r and parameter value r for each draw of zp-1i zp_1,r is then drawn from the distribution fl zp_1,o,r, & _1,r, vp-1). This process continues until z2,r is drawn and Jl = F1(zlo,r; 01,r vi) is computed with a commonly I

6.13 Monte Carlo Evaluation

163

available approximation routine for the univariate Student's t cdf. The simulated estimate p of p is then found as the sample average of the Jp values across the simulated sample of R elements 1

2

JP = R

1''1 (zpo,r; Qp

r, vv) JP-

r=1

vk)Jk_l for k = 2, ... , p-1. It is more efficient to estimate Jp by averaging over a large number of elements than to obtain close approximations of its components Jk for k < p. Therefore, a better estimate for p is where J k = F1(zko,r; Qk

r,

R

{fJFl(ko,r;&k2r,vk)}. r=1

k=1

The right-hand side of (6.36) remains unchanged if the integrand is divided or multiplied by any nonzero function of z. Let gv be a pdimensional pdf such that n

9v (z; v)

_

91 (zk;Tk) k=1

where g1 is a univariate pdf of a type to be mentioned below with Var(zk) = Tk = Qk'vk/(vk - 2), and Qk and vk are as defined above. Let G1(zk; Tk) be the associated cdf, and let z

91 (Zk; Tk G1 (40; Tk2)

Tk2)\

91C (Zk; zko,

be the conditional pdf. Finally, let n

9p (z; zo, v)

_ H 91C (Zk; zko, Tk) . k=1

With these definitions, one /can write p = Jv in terms of

A=

f

z,ko fl (Zk; Qk, yk) 00 91 (Zk; zko, Tk )

Jk-19i (Zk; zko, Tk) dzk

fl (Zk; k,2 vk) E9i

c

2

91(Zk; zko, Tk )

Jk-l

,

k = 2,...,p

and J1 = F(zlo;o'i,v1). Clearly, gP, and, more particularly, gi, is an important sampling density (see, for example, Hammersley and Handscomb, 1964). To evaluate p, the procedure is as follows: Generate random drawings z"p,r for r = 1, ... , R from the distribution gi (; zno Tn) i

Probability Integrals

164

compute the implied values zp_l,o,r and 'rP_1,r for each drawing of zr_i; draw 4-1,r from the distribution gi (; zp_l,o,r, 72_1 r, vp); and continue on until z2,0,r is drawn and Jl is computed. Based on this procedure, p may be written in the form 1

p=R

R

p 2

2 vk fl zk,r; 40,r, Qk,r

F'1 (z10,r;T1,r) r=1

k=1

gl (zk,r; zkQ,r,Tk r)

Three suitable choices for the importance density function are the logit with gl(x)

=

A q(1 T

- q),

where

q=

[1 +exp

(-Ax/T)]-1

and A = 7r/v'3-;

transformed beta (2, 2) density (Vijverberg, 1995) with

gi(x)

=

6x2(1 - z)2,

where z

=

exp (x/ci) 1 + exp (x/Q) '

the normal N(0, Q2) density. Vijverberg (1997, 2000) has developed a new family of simulators that extends the above research on the simulation of high-order probabilities. For instance, Vijverberg (2000) has reported that the gain in precision

using the new family translates into a 40% savings in computational time.

7

Probability Inequalities

Probability inequalities on Pr(Yi < y', Y 2 < Y 2 ,- .. , Yr < yr) for multivariate distributions have been a popular topic of investigation since the 1950s. It is well known (Khatri, 1967; Scott, 1967; Sidak, 1965, 1967) that, for arbitrary positive numbers yl, y2, ... , yr, the inequality p

Pr (IY1I < y,, IY2I < y2, ... , IYpI <_ yp)

>

11 Pr (IYkI <_ yk) k=1

holds for any random vector yT = (Y1, Y2, ... , Yp) having the multivari-

ate normal distribution with zero means and an arbitrary correlation matrix. A question then arises as to whether there is an analog of this for multivariate t distributions.

7.1 Dunnett and Sobel's Probability Inequalities Dunnett and Sobel (1955) obtained bounds for the probability integral

P = Pr(XI

<x1iX2 <x2,...,Xp <xp)

x1 > 0, x2 > 0, ..., xp > 0, when (X1i X2, ... , Xp) follows the central p-variate t distribution with degrees of freedom v and the correlation matrix R taking the special structure ri7 = bibs for all i $ j. Using the definition that X can be represented as (Z1/S, Z2/S, ... , Zp/S), where Z is a p-variate normal random vector with correlation matrix R and vS2 / i2 is an independent chi-squared random variable with degrees of freedom v, one can rewrite P as LP P = Pr SZl<xi,SZ2<x2i...,S

165

<xp}

Probability Inequalities

166

00

G (xls, x2s, ... , xps) h(s)ds,

L

(7.1)

where G is the joint cdf of Z and his the pdf of S. If Yo, Y1, Y2, ... , Yp are

independent standard normal random variables, then one can represent Zj = V 1 bjYo for j = 1, 2, ... , p. Using this result, one can rewrite G as

-bjYo <xjs,j = 1,2,...,p}

Pr{

G(xis,x2s,...,xps)

00 n 11 -°°j=1

z dz O()

1-b? ) (xjs+bjz'\

(7.2)

3

p

xjs + bjz

E

1-b32

where ¢ and t are, respectively, the pdf and the cdf of the standard normal distribution. Using the well known inequality

E

JP

p

> I E {Fj(x)}

Fj(x)

(7.3)

j=1

j=1

(where Fj denotes a cdf), one can now bound G by xjs + bjz

HE HE

G (xl s, x2s, ... , xps)

b

P

_

Pr

1-VYj-bjYo<xjs} J2

j=1 P

11 D (xj8) j=1

Substituting this result into (7.1) and applying (7.3) once more, one obtains the lower bound for P given by Dunnett and Sobel (1955) as

P>

T0O p

11 iP (xjs) f (s)ds j=1 P

f00

> 11 J j=1

(xjs) f (s) ds °°

7.1 Dunnett and Sobel's Probability Inequalities

167

P

lPr{Zj <xjs}

j=1 P

11 Pr {Xj < xj }

(7.4)

.

j=1

This lower bound for P holds more generally for any correlation matrix R with rij > 0 and any arbitrarily fixed.. (x1, . , xp). This is a conse-

quence of the fact that P is an increasing function of each rij for all i # j, while other correlations are held fixed. It can be shown further that P

Pr (X1 > x1 i ... , Xp > xp)

> H Pr (Xj > xj) j=1

and P

Pr (IX1 I < x1, ... , I XpI > xp)

11 Pr(IXjI>xj) j=1

Since the bound (7.4) does not depend on rij, it can be calculated easily from a table of the cdf of the univariate Student's t distribution. Dunnett and Sobel (1955) also obtained two sharper bounds by slight modifications of the above arguments: For even p > 2,

P>

p/2

11 Pr {X2j_1 < x2j_1, X2j < x2j} ,

(7.5)

j=1

and for odd p > 3,

P>

(p-1)/2

Pr {X1 < x1 } H Pr {X2j < x2j, X2j+1 < x2j+1 }

.

j=1

(7.6)

In the case where rij = p for all i 54 j and xj = x for all j, inequalities that are sharper than (7.5) and (7.6) can be obtained. Let 01(p)

= Pr(Xi
and

02(p) =( I X ,

d, .

. .

,

It is well known that /3k(p), k = 1, 2 are monotonically increasing in

Probability Inequalities

168 r23 .

(i j4 j) and, if r2j > 0, then Qk(p)

? #i(1),

k = 1, 2.

Tong (1970) provided the following sharper bounds for /3k /3k(1)}rf'z

fk(P) >- {0k(m)}p1m > /3ti(1) + {i3 (2) -

where p > m > 2. These inequalities certainly improve on (7.7), but neither of them are very sharp when p is large. Also observe that the first inequality in (7.8) depends on p and m only through their ratio p/m. Hence, for fairly large p and m as long as p/m is close to 1 (even if the difference p - m is not small), the first inequality is quite adequate. If p = 0, then a necessary and sufficient condition for /3k(p) -* /3'(1) for every fixed p is that v -> oo. Recently Seneta (1993) pointed out that the "sub-Markov" inequality /31(p)

>

[Pr {Xi < x, X2 < x}]p-1 / [Pr {X1 < x}]p-2

(7.9)

is sharper than the corresponding inequality /31(p)

>

[Pr {X1 < x, X2 < x}]p12

(7.10)

as given by (7.8). This fact is illustrated in the following table, which is taken from Seneta (1993).

Comparison of the bounds (7.10) and (7.9) for P. x chosen such that the true value of P = 0.95

p=3

V

x

10

2.34 2.24

15

20

60

p=9

2.19 2.10

10

2.81

15

2.67 2.60 2.48

20 60

Bound (7.10)

Bound (7.9)

0.945

0.946

0.946 0.945 0.944

0.947 0.946 0.945

0.921 0.924

0.921

0.926 0.934

0.927 0.936

0.924

7.2 Dunn's Probability Inequalities

169

Actually, (7.9) is a particular case of the following inequalities /31(p)

> Pr(Xj <x;j=1,...,m-1) x {Pr (Xm < x I Xj < x; j = 1, ... , m - 1)}r-'"+1 (7.11)

and

/32(P)

>

Pr(lXjl <x;j = 1,...,m-1) x {Pr (IXmI < x I 1Xjj < x; j = 1, ... , m - 1)}P_m+1

given by Glaz and Johnson (1984), who also provided a formal proof of the fact that (7.11) is sharper than (7.8). Dunnett (1989) wrote a Fortran programs for evaluating the integral (7.2). It uses Simpson's rule to compute an approximation to (7.2) in such a way that a prescribed accuracy is achieved. To approximate the integral of a function a(z), say, over an interval [a, b] using Simpson's rule, the value of the function is computed at the two end points and at the midpoint of the interval; then the approximate value of the integral is given by

{a(a)+4a1 a2

bounded

b

I +a(b)}

b

6

a,

by a4 (a, b) (b/-a) 5 /28)80, where a4 (a, b) is a bound with its error on the absolute value of the fourth derivative of a(z) over the interval (see, for example, page 66 in Shampine and Allen, 1973). The central processor unit time (on a VAX 8600 computer using single-precision arithmetic) taken to compute (7.2) ranges from 0.01 to 2.37 seconds for cases of equal correlation (r2j = p) and identical ranges of integration (xj = x). Slightly longer computing times are required for unequal correlations or different limits of integration. Dunnett (1989) suggested that his program can be used along with an appropriate numerical integration routine, such as the Integral Mathematical and Statistical Libraries'

(1987, Volume 1, Chapter 4) QDAGI, to evaluate the multivariate t probability integral (7.1).

7.2 Dunn's Probability Inequalities The univariate Student's t distribution has the property that the probability evaluated from -x to +x is an increasing function of the degrees of freedom v - this also applies to the probability from -oo to +x (see,

Probability Inequalities

170

for example, Ghosh, 1973, for details). Dunn (1965) pointed out that this monotonicity does not generalize to p dimensions in the usual multivariate t distribution. Specifically, let X have the central p-variate t distribution with degrees of freedom v and correlation matrix R. If F(x) is defined by

F(x)

= Pr {f1P=1 - x < Xk < X},

then F(x) equals the probability mass in the multivariate t distribution evaluated over a p-dimensional hypercube centered at the origin of the half side x and F(x) is the distribution of the maximum of the absolute values of the p X variables. Similarly, if G(x) is defined by G(x)

= Pr {f1P=1 - oo < Xk < x} ,

then G(x) equals the probability mass evaluated from -oo to x in each direction and G(x) is the distribution of the maximum of the p X variables. Dunn showed that, for any given x > 0 and degrees of freedom v1 > v2i there exists an integer K such that, for all p > K, Fp,v, (x)

<

Fp,vs (x)

Gp,,, (x)

<

Gp,, 2 (x).

and

Here, Fp,v and Gp,v are F and G as defined above, with dimension p and degrees of freedom v. This result covers the case of all correlations equal to 0. When all correlations are equal to 1, the distribution is the same as the univariate Student's t distribution, so that, for all dimensions, F(x) and G(x) are monotonically increasing functions of v. Other correlation matrices may be considered in some sense to lie between these two ex-

tremes. In various unpublished tables of F(x), the change is found to occur at a dimension where F(x) is approximately 0.25 or 0.30.

7.3 Halperin's Probability Inequalities Halperin (1967) extended the inequality (7.4) for generalized bivariate t distributions as follows. Let (Yi1, Yi2), i = 0, 1, 2, ... , r, r > 1 be independent samples from a bivariate normal distribution with zero means, variances o , 0'2, and covariances 0`1a2pi, pi 1< 1. Let Yi1, i = r + 1, ... , r + n and Yi2, i = r + 1, ... , r + m be independent normal samples with zero means and variances or1 and Q2, respectively, and I

7.4 Siddk's Probability Inequalities independent of (Yil, Y i 2 ) ,

171

i = 0 , 1 , 2, ... , r. Define

(X1, X2) =Y10 - ,Y20 - S2), S1 where r+n

EY2il

1

Si

=

r+

i-1

and

r+m

S2 =

2

r

+ m

Yi2 i-1

Halperin (1967) then showed that the probability integral of (X1, X2) satisfies the inequality

Pr(jXiI <x1,1X2j<x2) > Pr(IXiI <x1)Pr(jX2j <x2) for all real numbers xl and x2.

7.4 Sidak's Probability Inequalities In the bivariate case considered above, it is assumed that the correlation between Yil and Yi2 may be different for different i's. For a general p, but for a special correlation structure of Y's, Sidak (1967) established the following result. Let YT = (Y1, Y2, ... , Yp) have a p-variate normal

distribution with zero means and an arbitrary correlation matrix. Let ZT = (Zi1, Zit, ... , Zn,), i = 1, ... , n be a p-variate normal random sample, which is mutually independent and independent of Y, each of which has zero means, unit variances, and the decomposable correlation structure given by Corr (Zki, Zkj)

=

bibs

for i, j = 1,...,p; i # j; k = 1,...,n with 0 < bj <_ 1, i = ',...,PThen, IYil 711+...+Znl P

Pr

> i=1

2

< xi

IYZI

Zli +

Zn2

i

(7.12)

Probability Inequalities

172

Essentially the same result, assuming more generally only bi 1< 1, follows by an easy specialization of Corollary 8 in Khatri (1967). A general proof of the inequality (7.12) under the assumption that Y and all Zi's have the same normal distribution with zero means and an arbitrary covariance matrix was provided by Scott (1967). Unfortunately, this proof is correct only for p = 2, and Sidak (1971) produced a counterexample showing its incorrectness for p > 2. Sidak (1971) went on to show that, if I

Corr (Yi, Y-)

=

cicjrid

for i, j = 1,2,...,p; i Oj with I cj < 1 (j = 1,2,...,k) and {r2j} any fixed correlation matrix, and if Corr (Zii, Zip)

=

bciblj I

b1 1<1(i=1,2,...,p,

l = 1, 2, ... , n), then the left-hand side probability in (7.12) as a function

of cj is nonincreasing for -1 < cj < 0 and nondecreasing for 0 < cj < 1, so that it has a minimum for cj = 0 and, as a function of bli, is nonincreasing for -1 < bji < 0 and nondecreasing for 0 < b1i < 1, so that it has a minimum for b1i = 0. Hence, (7.12) is also true for this more general correlation structure. Sidak (1973) obtained an inequality using exchangeability when X is a central p-variate t random vector with the equicorrelation structure rid = p, i j. He showed that

Pr(b<Xi {Pr(b<Xl

{Pr (b < Xi < a)}P

for all p > r > 2 and a > b. In an earlier paper, Tong (1970) obtained similar results for a much larger class of random vectors.

7.5 Tong's Probability Inequalities We noted in Chapter 1 that the multivariate t density is Schur-concave in the particular case rid = p, i j. Tong (1982) used this property to derive certain probability inequalities. He showed that if f : RP -> [0, oo) is Borel-measureable and Schur-concave, then, provided that the integral exists, fA(X) f(y)dy is also a Schur-concave function of (x1,. .. , xp),

7.5 Tong's Probability Inequalities

where A(x) denotes the rectangular set

A(x) = {YIYER',IyjI <xj,i=1,...,p}. Taking f to be the pdf of the multivariate t, it follows that

Pr(IX;I < xj,j = 1,...,p) < Pr(jXjj < -t, j = 1,...,p),

where= -(xl++xv).

173

8

Percentage Points

From the 1950s numerous authors have tried to compute the percentage points of multivariate t distributions. It is an indication of the interest in problems leading to applications of this "new" distribution. This research continued well into the 1990s and is still going strong. Although some of the results have by now lost their practical importance - in our opinion - it is essential for historical reasons to describe a majority of these contributions. This will certainly assist historians and experts in multivariate distributions to gain a better perspective of the developments in the area. Moreover, many of the techniques involved in these calculations are ingenious and worthy of emulation and further investigation.

8.1 Dunnett and Sobel's Percentage Points Let (X1, X2) have the bivariate t distribution with joint pdf (6.1). For given v, p and probability level ry, let d denote the equicoordinate percentage point satisfying f (xl, x2; v, P)dxtdx2

=

-Y-

The value of d can be determined for any v by trial and error using (6.3) and (6.4). However, this procedure becomes more involved as v increases. Dunnett and Sobel (1954) derived an asymptotic expansion in powers of 1/v that expresses d in terms of the corresponding quantity e for the bivariate normal distribution: e is defined by e

re

f (xl, x2; oo, p) dxldx2 174

=

'Y

8.2 Krishnaiah and Armitage's Percentage Points

175

and can be obtained by interpolation in the classical tables of Pearson (1931), for example. Their expansion yields a good approximation of d even for moderately small values of v. The method of derivation is essentially the same as that used by Fisher (1941) in deriving an asymptotic expansion for the percentage points of the univariate Student's t distribution. Up to the terms of O(1/v), Dunnett and Sobel obtained

d = e-4v

e2+1-t't)

(t)

and by inverting

d+4 Jd2+1ll

=

e

,d) I

()

.

They also tabulated numerical values of the coefficients of the terms 1/vi for i = 0, 1, 2,3,4.

8.2 Krishnaiah and Armitage's Percentage Points Consider X having a central p-variate t distribution with degrees of freedom v and with the equicorrelation structure rid = p, i :/ j. Krishnaiah and Armitage (1966) evaluated the multivariate percentage point d by solving the integral P(d)

=f

d

...

fd

J

.f (xi, ... , xr; v, p)dxi ... dxp = 7

(8.1)

00

and produced extensive tables for all combinations of p = 1(1)10; v = 5(1)35; p = 0.05(0.05)0.9 and y = 0.90, 0.95, 0.975, 0.99 (see also Armitage and Krishnaiah, 1965). These computations use the approximation that xv-1

P(d)

(-x2) F exp (u/2)

2 foC

x

2

fo0

y

exp -y2)

v(1-P) dx + V 1-p

dydx,

is the cdf of the standard normal distribution and the upper limit c is chosen large enough to make the error of approximation as small as desired. Krishnaiah and Armitage (1966) took c = 10. where

Percentage Points

176

8.3 Gupta et al.'s Percentage Points Gupta et al. (1985) solved equation (8.1) by setting

IP(d) -

71

10-s

<

with P(d) computed by the approximation (6.14). The numerical evaluation of (6.14) involves the evaluation of the derivatives G(') (d) given by (6.15) for k = 0, 1, ... , 8. But it is easily seen from (7.2) that =

p

.

/ 00

1

G(k) (d)

pp

/2

lz (1:) Hk(z)0 (z)dz

(8.2)

where Hk(z) is the Hermite polynomial of degree k given by (6.11). By

letting A = rP_/ and B = 1/ and changing the variable by the transformation u = Az + Bd, (8.2) can be rrewriitten as G(k) (d)

A J°

=

(B)k k

(u Iu

A

d

Iu

I

dl

A

du.

Gupta et al. (1985) approximated this integral by j9A+4B

G(k) (d)

(B\ k

A J1

1

P(u)Hk I u A dJ `

(u

1 d I du,

\ A /

and the integration was carried out by Gauss' method over intervals of

length D = 0.5 starting from -9 until 9A + 4B was included. They provided tabulations of the percentage point d for all combinations of p = 1(1)9(2)19; v = 15(1)20, 24, 30, 36, 48, 60, 120, oo; y = 0.75, 0.9, 0.95, 0.99 and p = 0.1, 0.2(0.1)0.6;

p=

1(1)9(2)15; p = 0.7(0.1)0.9.

v = 15, 17, 20, 24, 36, 60, 120, oc; ry = 0.9, 0.95 and

8.4 Rausch and Horn's Percentage Points Rausch and Horn (1988) considered the particular case of (8.1) when the common p = 0. They used the approximation

-

n

P(d)

wi$P d

Zzi

i=1

The weights wi are calculated according to the formula Wi

=

I

zi 1

l

8.5 Hahn and Hendrickson's Percentage Points

177

where 3 = v/2 - 1,

Lm(z) =

lE

(:t)

(1! )c

are the Laguerre polynomials and z1i ... , zn are the zeros of L3 (z). Rausch and Horn computed d for all combinations of 3 < p < 100; 5 < v < 120, v = oc; and 0.5
8.5 Hahn and Hendrickson's Percentage Points Hahn and Hendrickson (1971) computed percentage points d by solving the equation

p(d) =

d

...

fd

fd

f

(X1,...

, xp> v, p)dxl ... dip = 'Y

(8.3)

d

fo r

all combinations of p = 1(1)6, 8, 10, 12, 15, 20; v = 3(1)12, 15, 20, 25, 30, 40, 60; p = 0, 0.2, 0.4, 0.5; and ry = 0.90, 0.95, 0.99. As one would expect, these values are comparable to the positive square root of the values given by Krishnaiah and Armitage (1966, Section 8.2). Hahn and Hendrickson's computations use the approximation that

P(d)

- -

,;;

f f.

4 ( dx+Id-g V1--

(-d

yl

P

0 (g) dg h(x)dx

where 0 and 4 are, respectively, the pdf and the cdf of the standard normal distribution and h is the pdf of xIlv.

8.6 Siotani's Percentage Points Siotani (1964) suggested two interesting approximations for computing d in (8.3). The first approximation is the value d1 satisfying

pPr (XZ > dl)

=

1 -'y;

this approximation had been suggested previously by Dunn (1958, 1961). By Bonferroni's inequalities, one notes that

1-ly-El(y,p)<1-P(d1)<1-7,

Percentage Points

178

where

Pr (XZ > dl, X? > d2) . i<j

Thus, if cl (ry, p) is sufficiently small, then one can use dl as a good estimate. A modified second approximation is the value d2 satisfying 2p Pr (XZ > d2)

=

1 - _Y + El ('y, p)

.

This time, one notes that -ez (,y, p) <7-P(d2) < e3 (7,p), where C2 (7, p)

_

Pr(Xz >d22 iX? >d2)-EI('Y,p) 2 i<j

and e3

(Y,p)

Pr (Xi > d2,Xj2 > d2,Xk > d2) .

_ i<j
Since both c2 ('Y, p) > 0 and e3 ('y, p) > 0, the absolute value of 'y - P(d2)

may be expected to be sufficiently small for the tail of the p-variate t distribution to correspond to 1- ry for values of 7 > 0.95. For the particular case p = 2, µi = µz = 0, and the equicorrelation structure rij = p, i # j, Siotani (1964) tabulated estimates of the probability in (8.3) for all combinations of d = 2.0(0.5)4.5; v = 10(2)50(5)90,100,120,150, 200, oo; and I p 1= 0.0(0.1)0.9, 0.95. He also illustrated applications to interval estimation of the parameters in the model of a randomized block design and for coefficients in a normal regression equation.

8.7 Graybill and Bowden's Percentage Points Graybill and Bowden (1967) derived bounds for d satisfying (8.3) for the special case p = 2 and p = 0. In this special case (8.3) becomes

Pr{X1
ry

or, equivalently,

Pr {max (Xi, X2)
= Pr {Xi + X2 < 2Fa,z,v-z }

8.7 Graybill and Bowden's Percentage Points

179

and

Pr {max (X,', X2) < F,,2,,-21 < Pr {X1 + X2 < 2F.y,2,v-2 }

< Pr {Max (Xi, X2) <2F-,2,_2}, where Fy,2,,,_2 is percentage point of the F distribution with degrees of freedom 2 and v - 2 corresponding to ry. Hence, one obtains F.y,2,,,_2 < d2 < 2Fy,2,v_2,

the bounds given by Graybill and Bowden. In a related development, McCann and Edwards (1996) obtained the following lower bound for the

left-hand side of (8.3) when the underlying correlation matrix R is of rank r 1'd

P(d)

1f0

miAFr-2 2

{2(ds)_2

1

r - 2-

2- l

+Fr-1,1 { (ds)r - 1

1

}

1 q(s)ds

(8.4)

with p-1

A=

arccos (rj,j+i) , k=1

where F,,,,,n is the cdf of an F distribution with degrees of freedom m and n, and q denotes the pdf of F, ,r/r. This inequality requires only the evaluation of a one-dimensional integral and depends on R through its rank r and also through the constant A. If one writes R = AAT for a p x r matrix A of rank r with rows a, , then it is interesting to note that the terms arccos(rj,j+1) in (8.5) are the angles between consecutive ak vectors, which are points on an r-dimensional sphere. It is also straightforward to show that the d that sets the right-hand side of (8.4) equal to ry is strictly increasing in A. This implies that, as A -+ oo, one has d rFl _ y r , , which is the percentage point given by Scheffe's method. On the other hand, as A -* 0, one has d -4 t(l_7)/2,v, the percentage point of the univariate Student's t distribution corresponding to (1 - y)/2. This is intuitively pleasing because A -4 0 implies that correlations in R approach 1, in which case the p-dimensional distribution becomes one-dimensional for all practical purposes.

Percentage Points

180

8.8 Pillai and Ramachandran's Percentage Points Pillai and Ramachandran (1954) tabulated solutions of (8.1) and (8.3) for

p = 1(1)8; v = 3(1)10, 12, 14(1)16(2)20, 24, 30, 40, 60, 120, 00; -y

0.05 and p

0;

p = 1(1)8; v = 5(5)20, 24, 30, 40, 60, 120, oo; ry = 0.05 and p = 0,

respectively. For computing these percentage points, Pillai and Ra-

machandran used the pdfs of

Up = max(X1,X2,...,Xp) and I Up I, which were derived as

f (up)

=

p(v/2) ,/2 P(v/2) 27r

00 1: k

k=O

(k+v+1)/2

6

up { (p + 2)u2 + 3v

}

k+v+1\ ap_1

xi'

2

Jk

and P /1

I\

pv

v/2

°O

iv+2k-1

V

(

6

(p+2k+v)/2

k=0

xI'

(-±_2 k+v1 by+2k-1 2

k

respectively, where the a'sfexp and b's are the coefficients of the expansions

(4) dt }exp-ky2)

{

[a(k)+a1k)y+a(k)y2+...] 6

and

respectively. Note that aok) = (1/2)k and bok-1) = 1.

8.9 Dunnett's Percentage Points Dunnett (1955) tabulated solutions of (8.1) and (8.3) for p = 1(1)9;

v = 5(1)20, 24, 30, 40, 60, 120, oo; 'y = 0.01, 0.05; and p = 0.5. For

8.10 Gupta and Sobel's Percentage Points

181

solving (8.1), Dunnett evaluated the integral in (7.1) by using tables of the multivariate normal cdf computed by the National Bureau of Standards. For (8.3), Dunnett bounded P(d) by P(d)

> [Pr (-d < X1 < d, -d < X2 < d)]P/2

(Dunnett and Sobel, 1955) and evaluated the probability integral of the bivariate t distribution using expressions (6.3)-(6.4). In a latter paper, Dunnett (1964) obtained approximations for d in (8.3) for all p's lying between 0 and 0.5.

8.10 Gupta and Sobel's Percentage Points Gupta and Sobel (1957) solved (8.1) for the special case p = 1/2. Note from (6.8) that

P(d) = Pr (Z < \d) and that Z = (Mr - Y)/S is asymptotically normal as both v and p tend to infinity. This allows for the use of a technique developed by Cornish and Fisher (1950) for computing the percentage points d directly without first computing a table of probability integral values. Applying their result, Gupta and Sobel arrived at

d = yy+a3Ic+a4Id+a3Ic2 +a5Ie+a3a4Icd+a321c3 + where y y is the percentage point of the standard normal distribution corresponding to y, ak; is the standardized cumulant defined in (6.12), and I, Id, II2, ... are tabulated in Table I of Cornish and Fisher (1950) for the probability levels y = 0.75, 0.90, 0.95, 0.975, 0.99, 0.995, 0.9975, 0.999 and 0.9995. Gupta and Sobel tabulated d for all combinations of p + 1 = 2, 5, 10(1)16, 18, 20(5)40, 50; v = 15(1)20, 24, 30, 36, 40, 48, 60, 80, 100, 120, 360, oo; and y = 0.75, 0.9, 0.95, 0.975, 0.99. Gupta and Sobel (1957) also obtained several bounds for the percentage point d satisfying (8.3). An upper bound for d is obtained by setting (6.8) to be equal to (1 +y)/2 while a lower bound is obtained by setting (6.8) to be equal to y. These bounds are best for large y's. For smaller values of y, Gupta and Sobel provided the following lower bound (1 + p)1/(2n)

d

- 4{F(1+)}11n

where xp 7 is the percentage point of the chi-squared distribution (with p degrees of freedom) corresponding to y.

Percentage Points

182

8.11 Chen's Percentage Points Chen (1979) provided an alternative formulation of percentage points d by solving the equation

F(d,...,d;p,v)-F(-d,...,-d;p,v) =

'y

(8.6)

for the special case p = 0, where

F(x,...x;p,v) =

x

f

...

fX00

f(Xi,...

Xr;v,P)dxP...dxl

and f is the joint pdf of a central p-variate t distribution with degrees of freedom v and the equicorrelation structure rzj = p, i j. Chen noted that (8.6) can be rewritten as 00

J where

the pdf of

Dk(-dy)}g(y)dy = ?,

(8.7)

is the cdf of the standard normal distribution and g denotes x,; further, by a change of variable, z = v/2y, (8.7)

becomes 2

f°°

k (,,12-dz

v

o

- 4) k

-/dz V/V

zi-1 eXp (-z2) dz F(v/2)

Using the fact that the tail integral

.tk \dz

2

- ,,Dk

-/dz

J1o 2

zi-1

eXp (-z2) dz F(v/2)

°° zv_1 exp (-z2) dz F(v12)

fo

7 10-8

for all v < 60, Chen approximated the left-hand side of (8.8) by P(d)

=

2

f fo

10

,,k

v dz

/v

- k -V2zv-1 exp (`z2) dz

v

F(v/2)

and found the value do such that P(do) = y. Tables of do were given for all combinations of 'Y = 0.8, 0.9, 0.95, 0.99; p = 2(1)20; v = 2(1)30(5)60;

andp=0.

8.12 Bowden and Graybill's Percentage Points

183

8.12 Bowden and Graybill's Percentage Points Bowden and Graybill (1966) presented percentage points of the bivariate t distribution when the percentage points are not necessarily equal, that is, the case where d

P(d, g)

g

= J J f (x1, x2) dx2dxl d

g

Setting D = d and A = g/d, one can rewrite

fj D

P(d, g)

=

AD AD

f (x1, x2) dx2dxl,

which can be solved for D using the expressions (6.3) and (6.4). Bowden and Graybill computed D for all combinations of p - 2 = 4 (2) 16 (4) 24 (6) 30 (10) 50; A = 0.5(0.1)1.5; p 1= 0.0(0.1)0.1(0.2)0.9; and ry = 0.90, 0.95. Trout and Chow (1972) extended this development for trivariate t distributions: Setting

JILl LI f (x1, x2, x3) dx3dx2dxl

=

It

and using the transformations D = d, A = g/d and B = h/d, the following expression is obtained AD

D

BD

AD-BD f (xl, x2, x3) dx3dx2dxl

=

-Y-

Tables for D were given for all combinations of v = 5(1)9(2)29; A = 0.5(0.1)1.5; B = 0.5(0.5)1.5; rii = 0.1(0.4)0.9; r12 = r13 = r23 = 0; and -y = 0.05.

8.13 Dunnett and Tamhane's Percentage Points More recently, Dunnett and Tamhane (1992) extended Bowden and Graybill's (1966) calculations for multivariate t distributions of any dimension with zero means and the equicorrelation structure ri.1 = p, i 54 j. Consider iid standard normal random variables Zj, j = 0,1, ... , p and let S be a Xv /v random variable independent of the Zj. Then the random variables defined by

X

i

=

1/1 -

Za

S

-

Z0

1 <_j
Percentage Points

184

have the desired multivariate t distribution. Thus

P = Pr(Xi < d1i...,Xp < dp)

f

00

00

J

Pr (Z1 <e1,. .. , Zp < ep) 0 (z) h(s)dzds, (8.9) 00

where ej = (djs + fp_z)l 1 - p, 0 is the pdf of the standard normal distribution and h is the pdf of S. Dunnett and Tamhane (1992) obtained the following recursive formula for evaluating 4),...p = Pr(Zi < e1,. .. , Zp < ep) in the integrand of (8.9) 4)1...p

Pr (Z1 < e2, Z2 < e3, ... , Zp_1 < ep)

(el )

+ Pr (Zl < e1i Z2 < e3, ... , Zp-1 < ep) x {-D (e2) - 4) (el)}

+ Pr(Z1 < e1i Z2 < e2,..., Zp_1 < ep_1)

x {b (ep) - 4) (ep_1)} ,

(8.10)

where denotes the cdf of the standard normal distribution. They also suggested the following algorithm for computing 4)l...p

Step 1: Calculate -Dj = D(ej), for j = 1, . . . , p, a total of p terms. for 1 < j < k < p, Step 2: Calculate 4 jk ='Dj'bk + Dj(4)k a total of (2) terms. Step 3: Calculate 4 jkl = 4kl4)j + Ijl((Pk - (Pi) + (Pjk(IDl - 4k), for 1 < i < j < k < p, a total of (3) terms.

Step p: Calculate

(D13...p((D2 - (b1) +

+

$l...p_1((Dp - $p_1).

The computational details of this algorithm can be found in Dunnett and Tamhane (1990). Kwong and Liu (2000) - using Kwong's (2001b) lemma - proposed the following modification of (8.10) M E(-1)j_1

j=1

Cmm

jJ/ {) (em-j+l)}3 Jm-j,

(8.11)

where m > 2, J1 = -t(e1), and Jo = 1 (see also Kwong, 2001a). There are three commonly known approaches for determining the percentage points d 1 , . . . , dp in (8.9) (after setting P = ry): the step-up procedure, the step-down procedure, and the simulation approach (Dunnett

8.13 Dunnett and Tamhane's Percentage Points

185

and Tamhane, 1995). We shall describe them below by means of recur< X(p) denote the sive algorithms. Throughout we shall let X(1) < order statistics of X1, . . . , X p and let c 1 , . . . , cp denote the corresponding ordering of d1,...,dp. Step-up procedure

(i) Take cl to be the 100'y percentage point of the univariate Student's t distribution with degrees of freedom v. (ii) Solve the equation Pr (X(l) < cl, X(2) < c2) Pr (Xl < c1i X2 < c2) + Pr (cl < X1 < c2i X2 < cl ) ry

for c2 by evaluating the two bivariate probabilities using (8.9) and the value for cl defined in (i). (iii) Solve the equation Pr (X(1) < cl, X(2) < c2i X(3) < c3)

= Pr (XI < c1i X2 < c2, X3 < c3) + Pr (Xl < c1, c2 < X2 < c3, X3 < c2) + Pr (cl < X1 < c2i X2 < c1, X3 < C3)

+ Pr (c1 < Xl < c2i c1 < X2 < c3, X3 < cl )

+Pr(c2 <X1
=

'Y

for c3 by evaluating the six trivariate probabilities using (8.9) and the values for cl and c2 defined in (i) and (ii), respectively. (iv) In general, the recursive formula below defines how the region over which the probability must be evaluated can be subdivided to obtain probability expressions

[X(1)
+ {cp_1 < Xl < cp, [X(2) < c1i ... , X(p) < cp_1] } , (8.12)

Percentage Points

186

< X(p) denote the order statistics of X2, ..., X p with X1 separated out. Formula (8.12) is applied recursively to the terms enclosed within the square brackets. This leads to a division of the region into p! subregions that have rectangular boundaries, making it possible to evaluate the individual probabilities (using (8.9)). where X(2) <

Step-down procedure: See Dunnett and Tamhane (1991) for a lucid description.

Simulation approach: This approach is feasible provided that p is not so large that sampling errors in the values of c3, ... , cp_1 accumulate and render the estimated value of cp too uncertain to be of practical use. The procedure for estimating cm given the values of cl, ... , cm_1 (Edwards and Berry, 1987) is as follows. (i) Let NT denote the total number of simulations to be performed

and choose NT so that No = (1 - y)(1 + NT) is an integer. (ii) Initialize a counter, NN = No.

(iii) For each simulation, draw m standard normal deviates Z1, .. . Z,,,, having the desired correlation structure and, if v is finite, a random x2/v variate S2.

(iv) Set Xi = Zi/S if v is finite or Xi = Zi if v = oo and order the < X(,,,,). X values to obtain the order statistics X(1) < (v) Check whether X(j) < cl, ... , X(m_1) < cr,,,_1. If this is the case, store the value of X(m) and return to step (iii). Otherwise, decrease NN by 1 and return to step (iii). (vi) After completing the NT simulations, find the estimate of cm by counting down NN from the top of the ordered values of the stored X(m). Note that this approach is general and does not impose restrictions on the correlation structure rid.

Dunnett and Tamhane (1992) computed values of dl, . . . , dp using the step-up procedure for all combinations of p = 2(1)8; v = 10, 20, 30, oo;

p = 0,0.1(0.2)0.5; and y = 0.95. Kwong and Liu (2000), using the same procedure but with the modification (8.11), computed values of dl, ... , dp for all combinations of p = 9(1)20; v = 10, 20, 30, oo; p = 0.1, 0.3, 0.5; and y = 0.95.

8.14 Kwong and Liu's Percentage Points

187

8.14 Kwong and Liu's Percentage Points In the case ri.7 = bibj, Kwong and Liu (2000) pointed out that (8.9) can be generalized to

P=

LI Hi...(z) h(s)dzds,

where H1...p = Pr(X1 < el, ... , Xp < ep), el = dlx and X3 are iid normal random variables with means bj z and variances 1- b?. The recursive formula (8.10) and the algorithm for computing it can be generalized in a natural manner. Hence the step-up, step-down, or the simulation-based procedure can be used to compute the percentage points d 1 , . . . , dp. A fourth procedure not discussed above is one based on approximation.

(i) cl and c2 can be determined as in the step-up procedure. (ii) To determine c3, replace r12, r13, and r23 by p3 = (r12 + r13 + r23)/3. Taking this as the common p and using the previous values c1, c2i apply the step-up procedure to estimate c3. (iii) To determine C4, replace 'r12, r13, r14, r23, r24, and r34 by fi4 (r12 + r13 +'r14 + r23 + r24 + r34)/6. Taking this as the common

p and using the previous values c1, c2i c3, apply the step-up procedure to estimate c4. (iv) In general, replace the (2) correlation coefficients by their arith-

metic average p. and use the previously calculated values of c1, ... , c,,,,to obtain an estimate for cm using the step-up procedure.

This procedure is similar to the ones presented in Dunnett (1985), Hochberg and Tamhane (1987, page 146), and Iyengar (1988).

8.15 Other Results Some other tabulations of percentage points of the multivariate t distributions with the equicorrelation structure are contained in the following references.

Paulson (1952) for p = 3,6 and p = 0. Dunnett and Sobel (1955) used the lower bounds (7.4), (7.5), (7.6), and (7.10) for p = 3, 9; v = 5, oo; p = 1/2; and 'y = 0.50, 0.75, 0.95, 0.99.

Halperin et al. (1955) for p = 3(1)10, 15, 20, 30, 40, 60; v = 3(1)10, 15, 20, 30, 40, 60, 120, oo; p = 1 - 1/p; and 'y = 0.95, 0.99.

Percentage Points

188

Gupta (1963) for p = 1(1)50; v = oo; p = 1/2; and y = 0.75, 0.9, 0.95, 0.975.

Milton (1963) for extensions of Gupta's (1963) tables for -y ranging from 0.5 to 0.9999. Steffens (1969b) for p = 2.

Dunn and Massey (1965) for p = 2, 6, 10, 20; v = 4, 10, 30, oo; p = 0.0(0.1)1.0; and ry = 0.5(0.1)0.9, 0.95, 0.975, 0.99. Tong (1970) for a procedure to calculate conservative estimates of the percentage points for p > 20 using tabulated values for p = 20.

Freeman and Kuzmack (1972) for p = 6, 8, 10(5)30; No = 10, 20, mean (40-70), 50, 100, mean (90-500), 500; p = 0; and ry = 0.90, 0.95, 0.99, where v = p(No - 1). Gupta et al. (1973) for p = 1(1)10(2)50; v = oo; p = 0.1, 0.125, 0.2, 1/3, 0.375, 0.4, 1/2, 0.64, 0.625, 2/3, 0.7, 0.75, 0.8, 0.875, 0.9; and y = 0.75, 0.9, 0.95, 0.975, 0.99.

Amos (1978) for p = 100; v = 100; p = 0.01; and ry = 0.05, 0.1, ..., 0.95.

Ahner and Passing (1983) for 1 < p< 20; 2 < v < 120, v = oo; p = 0; and ry = 0.95, 0.99.

Bechhofer and Dunnett (1988) for the most comprehensive table to date for p = 2(1)16, 18, 20; v = 2(1)30(5)50, 60(20)120, 200, oo; p = 0.0(0.1)0.9, 1/(1 + f); and ry = 0.80, 0.90, 0.95, 0.99. Kwong and Iglewicz (1996) for p = 4, 5; v = p(1)20, 24, 30(10)100,

120, oo; p = -1/(p - 1); and ry = 0.90, 0.95, 0.99 (note that the correlation matrix is singular).

There has been relatively little work concerned with the percentage points of multivariate t distributions when the correlations are not equicor-

related. Apart from those mentioned above, four results known to us are

In the case that the (i,j)th element of the inverse of R is

_

ria

2p/(p+ 1),

if i = j, -2/(p+ 1), if i i4j,

Goldberg and Levine (1946) computed the percentage point d satisfying d

00

rd

J-00

f (x1i... , xP; v) dxl ... dxr =

'Y

(8.13)

for combinations of p = 3; v = 1(1)30(3)60(15)120, 150, 300, 600, oo;

189

8.15 Other Results

and ry = 0.50, 0.75, 0.90, 0.95. This seems to be the earliest paper on this topic. In the case 2min(i,j){1-max(i,j)1

r2 =

p

l

1'

Freeman et al. (1967) computed the percentage point d satisfying (8.13) for combinations of p = 3(1)5; (v/p) + 1 = 10(10)100, 200, 500; and ry = 0.95. See also Bechhofer et al. (1954) and Table 4 in Dunnett and Sobel (1954).

In the case that the (i,j)th element of R is nine rz7

nj)'

(no + ni)(no +

where nk denotes some treatment sample size, Dutt et al. (1976) computed d in (8.13) for all combinations of y = 0.95, 0.99 and 3 < ni < 12, i = 0, 1, 2, 3 with p = 3 and the degrees of freedom v = E(ni - 1). See also Dutt et al. (1975). In the case rig = 0 for all i 54 j except that ni + ni+1

ni+1 -+n . + 2

for some treatment sample sizes nk, Lee and Spurrier (1995) provided

tables of one-sided and two-sided percentage points - of the form (8.13) - for 3 < p < 6 and nk = n (the balanced case). Liu et al. (2000) extended these tables for 3 < p < 10; v = 5(1)8(2)20, 25, 30, 40, 60, 120, oo; and ry = 0.90, 0.95, 0.99. See also Somerville et al. (2001).

Calculations of percentage points for singular correlation structures of the form rig = -bib; are discussed in Spurrier and Isham (1985) and Kwong and Iglewicz (1996). The former provided tabulations of the

percentage points for p = 3, 3 < nl <_ n2 < n3, 10 < N < 29, and ry = 0.90,0.95,0.99 when bk = nk/(N - nk) with N = nl + n2 + n3 (where nk denotes some treatment sample size). Calculations of percentage points for correlation structures more general than the decomposable structure ri7 = bibs are quite difficult and challenging. Even the generalization to quasi-decomposable structures of the form rig = bibs + ryij (Yang and Zhang, 1997), where the ryij's are nonzero deviations for some i and j, is a rather restrictive assumption. A solution is to find the "closest" R for a given matrix R, which still

190

Percentage Points

possesses the decomposable correlation structure (Hsu, 1992; Hsu and Nelson, 1998). One also has the choice of adopting the simulation approach or one of the general approaches due to Somerville (1997, 1998b) and Genz and Bretz (1999) - described in Section 6.10.

9

Sampling Distributions

Here, we shall consider sampling distributions of certain statistics associated with multivariate t distributions.

9.1 Wishart Matrix Suppose X,, ... , Xn is a random sample from a p-variate t distribution with the common pdf r '((v + p)/2) (rv)p/21' (v/2) r

IRI1/2

(v+p)/2

1

x I1+ L

- (xi V

µ)T R-1 (xi - µ)J

The joint pdf of the n independent observations is given by

f (xi,...,xn)

=f

(xl)...f (xn)

However, it is more instructive to consider dependent but uncorrelated t distributions. Joarder and Ahmed (1996) suggested the model F ((v + p)/2) F (v/2)

f (xi, ... xn) =

lxnv)p/2

x

1

IRIn/2

-(v+np)/2

n

R-1(xi-µ) i=1 (9.2)

which they referred to as the multivariate t model. Joarder and Ali (1997) remarked that this model can also be written as a scale mixture 191

Sampling Distributions

192

of multivariate normal distributions given by

f (xi,...,xn) n2

f

(2) n/2

exp

- 2 E (xi - µ)T (TZR)-1 (xi - µ) h(r) dT, Z-1

where T has the inverted gamma distribution with the pdf h (T)

(2/v)v/2I, (v/2) exp (\

=

Vl 2T2 I

(9.3)

for T > 0. Equivalently, Xj I T has the multivariate normal distribution Np(IL, r2R). Among others, Zellner (1976) and Sutradhar and Ali (1986) considered (9.2) in the context of stock market problems. By successive integration, one can show that the marginal distribution of Xj in the multivariate t model (9.2) is p-variate t. It also follows from (9.2) that E(Xj - µ) (Xi - µ) = 0 for j $ 1. Thus, in (9.2), although X1, ... , Xn are pairwise uncorrelated, they are not necessarily independent. More specifically, X1,. .. , Xn in (9.2) are not independent if v < oo, since independence would imply that X1,. .. , Xn are normally distributed. The case of independent normally distributed random vectors can be

included in (9.2) by letting v -4 oo. In the case v = 1, (9.2) is the multivariate Cauchy distribution for which neither the mean nor the variance exists. Kelejian and Prucha (1985) proved that (9.2) is better able to capture heavy-tailed behavior than an independent t model given by (9.1). The sampling quantities of interest are the mean vector and the sum of product matrix (Wishart matrix) given by 1

n

and n

A=

(Xi - X) (Xi - X) T

,

(9.4)

i-1

respectively. Sutradhar and Ali (1989) derived the corresponding pdfs, which are f (X)

=

v°/2r ((v + p)/2) IR/nI-1/2 jTp/2r (v/2) X

[v+(x-µ)TR-1(x-

9.1 Wishart Matrix

193

and

f(A) =

(( v + p)/2) (n-1)/2 A -(n-p-2)/2 vp/2I' (v/2) lTp (n/2) R I

trR-'A] -(v+p(n-1))/2

xw+

(9.5)

J

respectively, where A > 0, n > l+p and Fp(x) is the generalized gamma function defined by P

(2x -i+11 2

i=1

The distribution of the Wishart matrix, (9.5), has its applications in factor analysis. More specifically, in practice, one may be confronted with the situation where the observed data have a symmetrical distribution with tails that are fatter than that predicted by the normal distribution. In such cases, one could explicitly account for the observed "fat tails" by using the multivariate t model (9.2). Consequently, in factor analysis, analogous to the Wishart distribution, one may use the distribution of the sum of products matrix under (9.2). It is easily checked that, as v -4 oo, the pdf (9.5) converges to

f (A)

-

IRI-(n-1)/2IAI-(n-p-2)/2 2p(n-1)/2rp

((n - 1)/2)

1

exp

[_trR_1A],

which is the pdf of the usual Wishart distribution Wp(R,n). The pdf (9.5) can also be written as the mixture of distributions - n-1 2 TO IT2R'I IAI-(n-p-2)/2 f (A) 2P(n-1)/2rp ((n - 1)/2) x exp

[_tr ((T2R)-1 A)] f(r)dr,

where r has the inverted gamma pdf (9.3). This is equivalent to saying

that A I T = r2W has the usual Wishart distribution Wp(T2R, n). Joarder and Ali (1992) and Joarder (1998) derived various expectations of the Wishart matrix A. Specifically, one has the following expressions (A) n-1 R, E = 1

2/v

(n - 1)2R2 + (n - 1) {RtrR + R2} E (A2)

(1 - 2/v)(1 - 4/v)

v > k,

Sampling Distributions

194

_

E (IAIk)

IF (v/2 - kp) rp ((n - 1)/2 + k) IRIk v-kpr (v/2)

r((-1)/2)

v>k, _ E (IAIk A)

((n - 1)/2 + k)r (v/2 - kp - 1)

X

v-(l+kp)r (v/2) rp ((n - 1)/2 + k) IRIk R,

rp((n-1)/2)

n + 2k > 1,

217 (v/2 - kp + 1) Fp ((n - 1)/2 + k)

E (IAIk A-1)

E [(trA)2]

v > 2(kp + 1), IRIk R-1,

Ill-kp(n + 2k - p)I' (v/2) rp ((n - 1)/2) n + 2k > p + 2, v > 2(kp - 1),

_

(1

- 2/v) (1 - 4/v) v>4

[(n - 1) (trR)2 + 2tr (R2)]

,

and

=

E [tr (A2)]

n-1 [ntr (R2) + (trR)2] (1 - 2/v) (1 - 4/v)

,

v>4, where k is any real number and v > 0. These expectations are important tools in developing estimation theories for the correlation matrix, inverted correlation matrix, trace of the correlation matrix, and other characteristics of the correlation matrix, of the multivariate t model under quadratic loss functions. Extensions of these expectations to the class of scale mixtures of normal distributions - which may be useful in inferential works having a t distribution or the scale mixture of normal distributions as the parent population - are discussed below. Sutradhar and Ali (1989) derived an elementwise expression for the variance-covariance matrix of A. Letting mij denote the (i, j)th element of R'/2, they showed that Cov (Aij, Akt)

=

v2 _n

3(n - 1) E miumjumkumlu u=1

+(n - 1)(n - 2) EPmiumjumkumlu u=1

+(n - 1)2

miumjumkumlu u
9.1 Wishart Matrix

195

+(n - 1)2 E mivmjumkumlu u
+(n - 1) E miumjv (mkumly + mkvmlu) u
+(n - 1) E mivmju (mkumly + mkvmlu) u
P

u=1

u=1

(miumiu) (mkum1u

-(n - 1)2

for i#k, j 54 1,i,j,k,1=l,...,p,and _ Var (Aij)

=

P

v-2

P

2

+2(n-1)

(n - 1)2 t miumju

u=1

(U=1

+(n - 1)

(miumjv + miumju)

2

u
-(n - 1)2

miumju, u=1

)

for i, j = 1, ... , p.

Let A = TTT and A = SMST be the triangular and spectral decompositions of A. Let W = UUT be the triangular decomposition of W - W,(R, n). Let 11, ... 11P and ml, ... , MP denote the latent roots of W and A, respectively. Also define 1

n+p+1-2i and d`Z

=

v-2

1

v n+p+1-2i

with D = diag(dl,... , d,,) and D* = diag(di,... , dP). Then, some further expectation identities involving A useful in the estimation of R are E [log (IAI)]

= E [log (IWI)] + 2pE (log r) , P

E [log (IR-lAl)]

= >E [log i=1

(xn+i-i)] + 2pE (log -T)

,

Sampling Distributions

196

and

E [tr (R-'TATT)] =

v

v 2E [tr (R-lUAUT)]

(Joarder and Ali, 1997), where A is a positive definite diagonal matrix and r has the inverted gamma distribution given by (9.3). Joarder and Ahmed (1998) considered a generalization of the multivariate t model in (9.2) when the random sample X1, ... , Xn, is assumed to come from a p-variate elliptical distribution with the joint pdf 1 2

f (xi)

= Joy

exp H (xi - µ)T (T2R)-1 (xi - µ) } h(T)dr,

(R)n/2

where h(.) is the pdf of a nondiscrete random variable T. Many multivariate distributions having a constant pdf on the hyperellipse (x - µ)T R-1 (x - µ) = c2 may be generated by varying In this general case, the model corresponding to (9.2) has the joint pdf f (xi, ... , xn)

fC',

(xi - µ)T (7-2R)

2

(27r)I p/22 exp {

l

_1

(xi - µ) h(T)dr.

Z_1

(9.7)

The observations X1, ... , Xn are independent only if T is degenerate at the point unity, in which case the joint pdf (9.7) denotes the pdf of the product of n independent p-variate normal distributions each being Np(µ, R). Furthermore, if v/T2 has the chi-squared distribution with degrees of freedom v, then (9.7) reduces to (9.2). The pdf of the Wishart matrix A under the generalized model (9.7) takes the form

n-

n/2

f (A)

1

IT2R2np/2rA(n/2)-1)/2

exp {_tr 2((T2R)-1 A)] h(T)dr,

=

where A > 0, n > 1 +p, and rp(-) is as defined in (9.6). Some expectations of A useful for estimating R are IR"Ir

2pr

r rn(n l )r)

E (JAI') E [IAA

k A]

=

2

kp

(n + 2k)

rp (n/2 + k)

r (n/2)

Y2pr,

k

IRS R'Y2kp+2,

and

E [(trA)2]

=

nry4

[n (trR)2 + 2tr (R2)]

,

9.1 Wishart Matrix

197

where 'y E R, k E R, n + 2k > 0, 'y2pr = E(7- 2pr) > 0, 'Y2kp+2 = E(T2kp+2) > 0, and ry4 = E(T4) (all assumed to exist).

The pdfs of A in the real and complex cases - under the independence model (9.1) - were originally studied by Cornish (1955) and Gupta (1964), respectively. Nagarsenker (1975) provided a very detailed study

of the distribution of A and its quadratic forms. He investigated both the noncentral real and the noncentral complex cases. Let Y be a p x n matrix of iid normal random variables with means E(Yij) = pij and covariance matrix a2R. Assume that S is an indepen-

distribution. Then the

dent random variable having the noncentral version of A is defined by

=

aij

l

nn`

(Yjk-Yj),

Q2E(Yik-Yz) k=1

where 1

Y

n

n 1Yj. j-1

In the real case, S2A has the noncentral Wishart distribution. In the complex case, it should be interpreted as a Hermitian positive definite matrix having the noncentral complex Wishart distribution (James, 1964). In the noncentral real case, Nagarsenker (1975) established that

the pdf of A is given by the complicated expression involving zonal polynomials

f(A) (aF)p(n-1) I R I-(n-1)/2I A I(n-p)/2

exp {-trR-1/z, T/ (2a2) } (2v)P(n-1)/2rp ((n - 1)/2) r(v)

Up(n-1) (or )2k F (v

0

+ k + p(n - 1)/2) Ck (R-lµ,LT R-1A) k!((n - 1)/2),, (4vQ4)k {1 + (Q'2trR 1A) /2vQ2}

X

k=O k

+p(n2

1)

+ k,

(9.8)

where ,={kl,...,km}, k1>k2>...>k,,,,>0,k1+k2+ (x)

r(r-1)/4

-

+ km = k,

rP (X, K) )

-

1

...

(x+r -

2

Sampling Distributions

198

and Ck(T) are symmetric homogeneous polynomials of degree k in the

latent roots of T. In the particular case a = a' and it = 0, (9.8) reduces to the expression given in Cornish (1955). If B is an (n - 1) x (n - 1) symmetric positive definite matrix of full rank, then Nagarsenker (1975) established further that the quadratic form Q = ABA' has the formidable pdf

f(Q) (0.l)P(n-1)

I R I-(n-1)/2I

Q

I(n-p)/2

cP(n-1) (2v)P(n-1)/2rp ((n - 1)/2) r(v) I B JP/2 X 00

C,, (-R-1Q/2v) C, (B-1) r (p(n - 1)

k=O c

k! (Q/Q')2k Ck (In-1)

+k+v

2 (9.9)

In the particular case o, = ci' and B = In-1, (9.9) reduces to equation (14) in Cornish (1955). Nagarsenker also provided the joint cdf and the moment generating function of Q as well as the corresponding expressions for the noncentral complex case (which generalizes those given in Gupta, 1964).

9.2 Multivariate t Statistic

A random variable X with iid copies X,, X2, ... is said to be in the domain of attraction of the normal law if there exists an -4 oo such that n

-I: 1

-+

an i=,

N(0,1)

as n -4 oo. It is well known that, for X in the domain of attraction of the normal law, the t statistic defined by _A) E 1 (Xi Tn == -+ N(0,1) (9.10) En

I (Xi -

X)2

Ei

as n -> oo, where, as usual, (1/n) 1 Xi. Sepanski (1994, 1996) provided two multivariate analogs f (9.10). Let

X be a p-variate random vector with mean vector µ and covariance matrix E. Also let X1, X2, ... be iid copies of X. Then X is said to be in the domain of attraction of a p-variate normal law if there exists

9.3 Hotelling's T2 Statistic

199

an -+ oo such that

1 a

(X i - N-)

-* N (0, C)

(9.11)

n i=1

for some nonsingular matrix C. Sepanski (1994) defined the multivariate

t statistic by n

Tn = Dn 1/2

(Xi - µ-)

,

(9.12)

i=1

where, for some sequence bn > 0, Dn = Cn + bnI and n

Cn =

(Xi - X) (Xi - X)T. i=1

Note that Cn is symmetric nonnegative definite while Dn is symmetric positive definite. Under the assumption that X satisfies (9.11), Sepanski (1994) showed that Tn -> N(0, I) as n -4 oo. Sepanski (1996) established the same limiting result under weaker conditions by taking n

Tn = Cn1/2 E (Xi - µ)

(9.13)

i=1

and considering its behavior when X is in the generalized domain of attraction of a normal law, which means that there exist matrices An and vectors µn such that n

Xi - µn -4 N (0, I)

An

(9.14)

i=1

as n -* oo. See Hahn and Klass (1980a, 1980b) for several examples of random vectors satisfying this condition and for an algorithm for constructing the normalizing matrices A.

9.3 Hotelling's T2 Statistic A customary approach to the estimation/testing problem is based on the so-called Hotelling's T2 statistic. It is defined by H,

= n2 (X - µ)T Cn 1 (X - µ)

Under the normality, it is well known that (n - p)Hn/(p(n - 1)) is distributed as an F distribution with degrees of freedom p and n - p (see, for example, Anderson, 1984). The distribution of H,2, has been studied

Sampling Distributions

200

under a mixture of two normal distributions by Srivastava and Awan (1982) and Kabe and Gupta (1990). Iwashita (1997) investigated the asymptotics of Hn under an elliptical distribution. Unfortunately, there are no direct results for the specific case of multivariate t distributions. For completeness, however, we shall survey the results when X has an elliptical distribution. In this case, the characteristic function of X can be written as

0(t) = exp {itT m} I (mTQ-lm)

(9.15)

(Kelker, 1970) and the parameter

for some nonnegative function

11

'F

_

(0) {W'(0)}2

- 1,

which controls the kurtosis of the distribution. Iwashita (1997) provided an asymptotic distribution of H.2 under the null hypothesis that m = µ and a local alternative of it. Up to the order of 1/n, the asymptotic null pdf of Hn is given by 2

f (x)

=

gp(x) +

E ojgp+2j (x) j=0

+0

(n)'

where CO

-Ip{p+k(p+2)},

=

-2p{1 -r.(p+2)1,

Cl

=

c2

I

p(p + 2) (1 - n),

and denotes the pdf of a chi-squared distribution with degrees of freedom k. Iwashita (1997) also derived the percentiles and approximate powers of the Hn statistic. An asymptotic expansion of the cdf of Hn under the two assumptions (i)

E(II Y 118) < oo, where Y = E-112(X - µ) and X is a p x 1

random vector with mean vector µ and covariance matrix E; (ii) the distribution of Y = (Y1, ... , Yp) has an absolutely continuous component with a positive density on some nonempty open set U such that 1, y1, . on U

.

.

, yp, yi , YlY2, ... , yep are linearly independent

9.3 Hotelling's T2 Statistic

201

is given by Fujikoshi (1997). It takes the form 3

= Gp(x) + 1 E NjGp+2j (x) + o I 1n

Pr (H,2, < x)

I

(9.16)

j=0

uniformly for all positive real numbers x, where Gk(.) denotes the cdf of a chi-squared distribution with degrees of freedom k. The coefficients /3j's are given by 6(r( 1)) 2

_ 4p2 +

=

00

1 _ 2p

N1

K41) 4

3

3((1))2 + 2 k4(1)

1

1

2

2

02

= 4p(p+2) - 2 (K32))

__4

X4

1)'

and

(,(2) )2' /33

3

3 (x(1))2 +2

3

where K31)

= J: (X(Zj,k))2, i,j,k

K(2)

K(i,i,k)/£(j,k,k)'

i,j,k X41)

=

K(i,i,j,j)

i,j

are the jth cumulants of Y. If X has the elliptical distriand bution given by (9.15), then a3 vanishes to zero and /3k reduces to ck for k = 0, 1, 2. Kano (1995) obtained the same asymptotic expansion as in (9.16), using a different method. It is well known that, for large samples, H,2, has a limiting chi-squared distribution with degrees of freedom p. The usual underlying assumption for this result is simply that E 11 X j1< oo. More general limiting behavior of H,2, has been studied by Eaton and Efron (1970), Sepanski (1994), and Fujikoshi (1997). Sepanski (1994) showed that, under the

Sampling Distributions

202

assumption of the generalized domain of attraction (defined in (9.14)), the modified Hotelling's T2 statistic Hn2

= n2 (X - µ)T Dnl (X - µ)

still has an asymptotic chi-squared distribution with degrees of freedom p. Eaton and Efron (1970) studied the distribution Hn when X has

orthant symmetry, that is, X has the same distribution as DX for any choice of the diagonal matrix D with diagonal elements equal to ±1. We shall now consider the Hotelling's T2 statistic in the context of testing equality of means. Suppose X1 = (X11, ... , X lnl )T and X2 = (X21, ... , X2n2)T are two samples of size n1 and n2, respectively. In analogy with (9.2), assume that X1 and X2 have the joint pdf given by f (x1, x2)

a

JRI

-n/2

-(v+np)/2

n:

2

v-2+E

(xij - µi)T R1 (xij - µi)

i=1 j=1

where n = n1 + n2. It is immediate that Xij is p-variate t with mean vector µi, correlation matrix R, and degrees of freedom v. Also, the elements of the combined sample of size n = n1 + n2 are pairwise uncorrelated. The Hotelling's T2 for testing equality of means takes the form (X1 T2

nn+n2 (X1

- X2)T spooled

- X2)

,

where n,

1

Spooled

=

2

nl +n2 - 2

E z=1 j=1

(Xi.7 - Xi)

T T.

Sutradhar (1990) derived the nonnull distribution of the T2 statistic, given by the pdf f (t2)

r v 2 2

I #6/(,-2) (r + 1, 2- 1)

nl+n2-p-1 XOf r+ p 2 2' where

I'(k)r(m)xk-1 r(k + m)(1 + x)k+m

'

(9.17)

9.3 Hotelling's T2 Statistic

203

and

nnin2 n2

R-1 (i1 - P2)

(µ1 -

Note that, under Ho : it, = µ2, where S = 0, the pdf of T2 in (9.17) reduces to

f(t2)

= of

n1+n2-p-1 (2)

2

which implies that, under Ho, T2(n1 + n2 - p - 1)/p has the usual F distribution with degrees of freedom p and n1 + n2 - p -1. Thus the null distribution remains the same as in the normal case. Furthermore, the power of the Hotelling's T2 test can be computed by using the nonnull pdf in (9.17). Kozumi (1994) considered testing equality of means when the two samples X1 and X2 have mutually independent t distributions with equal correlation matrices and equal degrees of freedom. When the sample sizes are equal (say, n1 = n2 = n) the T2 statistic is given by Td = nyTShcy,

where y and Sd are, respectively, the sample mean and the sample covariance matrix of the differences yj = xlj - x2j. For unequal sample sizes, assuming without loss of generality that n1 < n2i the T2 statistic is given by

TS = n12TS312, where nl Zj

=

Xlj

1 X 2j j +

V

n2

/-n -1n-2 1

n2

X2s -

n2 1

1

X2s 1

and z and S8 are the sample mean and the sample covariance matrix of the zj's. It should be noted that Ts reduces to Td in the case n1 = n2 = n. Under the Ho : µ1 = z2, (n1 - p)TS /(p(n1 - 1)) has the usual F distribution with degrees of freedom p and n1 - p. The nonnull pdf of T3 is given by the infinite sum involving Student's t pdfs

f (t8)

(n1V ( 1) F2

x

(v/2) 2 8

-

k!B (p/2 + k,((n1

p/2+k-1

n1-1

xJ(x,n1,n2,k,6,v),

t2 1

+

8

n1-1

v p)/2) -(ni/2+k)

Sampling Distributions

204

where 6 = (µ1

- µ2)T R-1(µl - 12) and the integral 1

J(x;nlin2,k,6,v)

L

{x(1 - x)}k+v/2-1 {n26x(l - x)

-(k+v)

n2 ) x+vn2

+V

ni

\\

dx.

ni

Kozumi (1994) also provided an expression for the cdf of Ts and calculated the powers of TS corresponding to the sizes a = 0.01, 0.05, p = 5 and for various values of n1i n2, 6, and v.

9.4 Entropy and Kullback-Leibler Number The forms of entropy and Kullback-Leibler number for the multivariate t distribution were discussed earlier in Chapter 1 (see equations (1.27), (1.29), and (1.31)). Here, we shall discuss the corresponding sampling properties. The entropy for the central p-variate t involves the correlation matrix

R, and it is known that the maximum likelihood estimator of R for a sample of n observations is based on the Wishart matrix A in (9.4). Hence it is of interest to consider the sampling properties of the difference 6 = H(X; A) - H(X; R). Guerrero-Cusumano (1996a) derived the corresponding moment generating function, mean, variance, and some asymptotics. Specifically,

M(t) =

vPt/2rn-l+t)r(v-tp) t\

2

1

2

P

n -i

2

rP(n-'-)rrl 2

(vl

\2

p21+21ogv'

2

i=1

Var (6)

4 i-1

2

and

E 2

v S*

as n -a oo, where

=

n

-4

N(0, p)

2

v [2, - p log {v(p - v) j]

denotes the digamma function.

Note that

9.4 Entropy and Kullback-Leibler Number

205

Hu(X; R) = H(X; A) - E(5) is an unbiased estimator for H(X; R) with E(H,,,) = 0 and Var(Hv,) = Var(8). Also note that, as v - oo, (n_2

E(8)

2E z-1

2) +plog2,

coinciding with the result given in Ahmed and Gokhale (1989) for the multivariate normal distribution. The expression for Var(S) given above is also valid for the multivariate normal distribution since it is independent of v. The Kullback-Leibler number for the central p-variate t is given by (1.31). The corresponding maximum likelihood estimator for a sample of n observations is

/,/-2) nAll-2log(vT

T(X;R) = S2--log

the sampling quantity of interest is the difference S = T(X; R) T(X; R). Guerrero-Cusumano (1996b, 1998) derived the corresponding moment generating function, cumulant generating function, cumulants, mean, variance, and some asymptotics. Specifically,

Mt O = (J) Ptr2rP1(n-1+t)r(v+tpl \ 2 J 2 J FP(n21)I'\2/ K(t)

2tlog(v-2)+logr(v 2pt)-logr\21 +E{logrn-2 i-t)

l

K3

pj-"bu-)

=2 E(S) =

Var (6)

p)

(v

2

PV)(1)

{

J' i)

2

2

2\v21

4

n2 -i

(n

l np

1

=

log I'

n-1

+21og(v-2)

(v+v) -,(1) (n 2 i }

206

Sampling Distributions

and

V +P6 P

_+ XY+p)2

as n -> oo and v -> oo. Furthermore,

(n-1)T(X;R)

-4

X2p(p-1)/2

and

n-16

N

O,tr(B2) -p 2

as n -* oo, where B = Ad 1/2AAd 1/2 with Ad denoting the diagonal matrix of A. In the latter limit, it is assumed that v is known. When v is unknown, the limit still holds for n sufficiently large. The exact distribution of 6 is quite complicated to obtain in a closed form.

10 Estimation

The material in this chapter is of special interest to researchers attempting to model various phenomena based on multivariate t distributions. We shall start with a popular result in the bivariate case.

10.1 Tiku and Kambo's Estimation Procedure In Chapter 4, we studied a bivariate t distribution due to Tiku and Kambo (1992) given by the joint pdf {1+(x2 - µ2)z

1

f (X1, X2)

=

aio2

ko2

k (1 - p2) 1

x exp

2o, (1- p2)

Ql

xi - Al - p2 (x2 - µ2) z (10.1)

Here, we discuss estimation of the parameters p1i 42i U1, a2, and p when v is known. The method for estimating the location and scale parameters

developed by Tiku and Suresh (1992) is used for this problem. For a random sample {(X1i, X2i), i = 1,... , n} from (10.1), the likelihood function is L

oc

{ai o (1

n/2

- p2) }

n

j

{ 1 + X(z:i) - µz 1

1

2Q 1 ( 1

-

-V

k0,2

P2) i=1

{Xii:i - P1 -

}2] z

(X(z:i) - µ2)

where k = 2v - 3, X(2:i), i = 1, ... , n are the order statistics of X2i and X[l:i], i = 1, ... , n are the corresponding concomitant X1 observations. Consider the following three situations: 207

Estimation

208

(i) Complete samples are available and v is not too small (v > 3). (ii) Complete samples are available but v is small (v < 3). (iii) A few smallest or a few largest X2i observations and the corresponding concomitant X[l:i] are censored due to the constraints of an experiment. This situation arises in numerous practical situations. In a time mortality experiment, for example, n mice are inoculated with a uniform culture of human tuberculosis. What

is recorded is X2i: the time to death of the first A(< n) mice, and Xli: the corresponding weights at the time of death. These situations also arise in the context of ranking and selection (David,

1982). We provide some details of the inference for situation (i) as described in Tiku and Kambo (1992). Using a linear approximation of the likelihood based on the expected values of order statistics, it is shown that the maximum likelihood estimators are

Ql

xl--- (x-µ2)i

=

µl

Q2

=

8

1+

= x2 -

µ2

S2 12

2

2 \Qs2

- 1)

s2 2

Pa2 (x1

-'al),

1

Q2

2 +2 I 2

= uu

S1

- 1J

\S1

and

P=

s12 Q2

2 all

where (xl, x2) are the usual sample means, (s2, s2) are the usual sample variances, and s12 is the sample covariance. The estimators µl, µ2, al, 62 i and p are asymptotically unbiased and minimum variance bound estimators. The estimator Qi is always real and positive while the es-

timator p always assumes values between -1 and 1. The asymptotic variances and covariances of the estimators can be written as

V

( \\\

Vl 0

0

I

V2/'

10.1 Tiku and Kambo's Estimation Procedure

209

where

vi

paia2

Pa2ai

a2

1

vl =

n

2mv - nk ( P2aia2 Pal a2 2vmna2 \ Pat orl a2

/

(10.2)

is positive definite and is the asymptotic variance-covariance matrix of (µi, µ2) while pal a2

a2

Pai (l- p2) Pat (1 - P )

v2

Poll (1 - p2)

pa2 (1 - p2)

2 (1 - p2)2

Pa1 s 2

a21

1

=2n

8

2n(2 + 6)

P4ai p2ala2

P2aia2

P3a1 (1 - p2)

pa2 (1 - p2)

a2

p2)

P3a1 (1 p92 (1 - p2)

p2 (1 - p2)2 ) (10.3)

is positive definite and is the asymptotic variance-covariance matrix of (vl, &2i p). The parameters m and 6 are determined by the linear approximation of the likelihood. Interestingly, Var(µl) and Var(µ2) decrease with increasing p2 unless v = oo. The first component on the right of (10.2) is the variance-covariance matrix of µ1, and I2 under bivariate normality, and the first component on the right of (10.3) is the asymptotic variance-covariance matrix of a,, Q2, and p under bivariate normality. The second components in (10.2) and (10.3) represent the effect of nonnormality due to the family (10.1). The asymptotic distri-

bution of J(µl -µl,µ2 - lie) is bivariate normal with zero means and variance-covariance matrix nVl. For testing Ho : (µl,µ2) = (0, 0) versus H1 : (µl,112) 54 (0, 0), a useful statistic is T _ (µl, µ2)T Vl (µl,µ2), the asymptotic null distribution of which is chi-squared with degrees of freedom 2. The asymptotic nonnull distribution is noncentral chisquared with degrees of freedom 2 and noncentrality parameter AV

where A,,.

=

n 1

= a- +

(2kmvn

- 1)

)2' \(92

2

{()2(2(\\f\ (0'2)(L-) 'al a1

Note that A is the noncentrality parameter of the asymptotic nonnull distribution of the Hotelling's T2 statistic based on the sample means

Estimation

210

(ii, x2), sample variances (32, s2), and the sample correlation coefficient 812/(8182). Tiku and Kambo (1992) also provided evidence to the fact that the use of T2 in place of the Hotelling's T2 statistic can result in a substantial gain in power.

10.2 ML Estimation via EM Algorithm Consider fitting a p-variate t distribution to data x1,.. . , x,, with the log-likelihood function

L (µ, R, v)

9 log jRI

-v

p E log (v + si) ,

(10.4)

i=1

where si = (x - µ)TR-1(x - µ) and v is assumed to be fixed. Differentiating (10.4) with respect to µ and R leads to the estimating equations

µ = ave{wixi}/ave{wi}

(10.5)

R = ave{wi(x-µ)(x-µ)T},

(10.6)

and

where wi = (v + p)/(v + si) and "ave" stands for the arithmetic average over i = 1,2,. .. , n. Note that equations (10.5)-(10.6) can be viewed as an adaptively weighted sample mean and sample covariance matrix where the weights depend on the Mahalanobis distance between

xi and µ. The weight function w(s) = (v + p)/(v + s), where s = (x - µ)'R-'(x - µ), is a decreasing function of s, so that the outlying observations are downweighted. Maronna (1976) proved, under general assumptions, the existence, uniqueness, consistency, and asymptotic normality of the solutions of (10.5)-(10.6). For instance, if there exists a > 0 such that, for every hyperplane H, Pr(H) < p/(v + p) - a, then (10.5)-(10.6) has a unique solution. Also, every solution satisfies the consistency property that R) = (µ, R) with probability 1.

The standard approach for solving (10.5)-(10.6) for µ and R is the popular EM algorithm because of its simplicity and stable convergence (Dempster et al., 1977; Wu, 1983). The EM algorithm takes the form of iterative updates of (10.5)-(10.6), using the current estimates of µ and R to generate the weights. The iterations take the form ave {w(m)xi} /ave {w(m) }

10.2 ML Estimation via EM Algorithm

211

and

R(m+1)

= ave { w(m) (xi

- µ(m+1)) (xi

µ(m+1))TI,

-

where w(m) z

_ (v + p)

(v + (x2 - µ(m))T (R(m))

(xi

l

- µ(m))Jf

This is known as the direct EM algorithm and is valid for any v > 0. For details of this algorithm see the pioneering papers of Dempster et al. (1977, 1980), Rubin (1983), and Little and Rubin (1987). Several variants of the above have been proposed in the literature, as summarized in the table below.

Algorithm

Extended EM Restricted EM MC-ECM1 MC-ECM2 ECME1 ECME2 ECME3 ECME4 ECME5 PXEM

Primary References

Kent et al. (1994), Arsian et al. (1995) Arsian et al. (1995) Liu and Rubin (1995) Liu and Rubin (1995), Meng and van Dyk (1997) Liu and Rubin (1995), Liu (1997) Liu and Rubin (1995) Liu and Rubin (1995), Meng and van Dyk (1997) Liu and Rubin (1995), Liu (1997) Liu (1997) Liu et al. (1998)

Consider the maximum likelihood (ML) estimation for a g-component mixture of t distributions given by g

f (x;

E 7rif (xi l-L4, Ri, vi)

,

i=1

where

f (x; tti, Ri, vi)

r'((vi+p)/2) (irvi)p/2 r (vi/2)

IRiI1/2

T X 1+ (x - Ai)Ri (x-N'i) 1

vi

-(v:+p)/2

Estimation

212

R1)T,.. ., (µg, R9))", and v = = (in , ... , 7fg-1, 0T , VT) T, 0 = ((pr, (v1, ... , v9)T . The application of the EM algorithm for this model in a ,p

clustering context has been considered by McLachlan and Peel (1998) and Peel and McLachlan (2000). The iteration updates now take the form n

m

TZj

j=1

m

Uij

In xj

j=1

m Tzj

m

uzj

and R(m+1)

_

E j=1

(7)U gym) (xj _ µ(m+1)) (x7 -µ2m+1)1 T J l

J`

rim), j=1

where u(in) aj

_

lT

1

(xj

(xj

RZm)

v( m) +

- µ2m)/

and

TZj

=

(m') f (xj I-RZm), RZm), v(m))

f (xj; F(m)) The EMMIX program of McLachlan et al. (1999) for the fitting of normal mixture models has an option that implements the above procedure for the fitting of mixtures of t-components. The program automatically

generates a selection of starting values for the fitting if they are not provided by the user. The user only has to provide the data set, the restrictions on the component-covariance matrices (equal, unequal, diagonal), the extent of the selection of the initial groupings to be used to determine the starting values, and the number of components that are to be fitted. The program is available from the software archive StatLib or from Professor Peel's homepage at the Web site address http://www.maths.uq.edu.au/-gjm/

10.3 Missing Data Imputation When a data set contains missing values, multiple imputation for missing data (Rubin, 1987) appears to be an ideal technique. Most importantly,

it allows for valid statistical inferences. In contrast, any single imputation method, such as filling in the missing values with either their

10.3 Missing Data Imputation

213

marginal means or their predicted values from linear regression, typically leads to biased estimates of parameters and thereby often to an invalid inference (Rubin, 1987, pages 11-15).

The multivariate normal distribution has been a popular statistical model in practice for rectangular continuous data sets. To impute the missing values in an incomplete normal data set, Rubin and Schafer (1990) (see also Schafer, 1997, and Liu, 1993) proposed an efficient method, called monotone data augmentation (MDA), and implemented it using the factorized likelihood approach. A more efficient technique to implement the MDA than the factorized likelihood approach is provided by Liu (1993) using Bartlett's decomposition, which is the extension of the Bayesian version of Bartlett's decomposition of the Wishart distribution with complete rectangular normal data to the case with monotone ignorable missing data.

When a rectangular continuous data set appears to have.longer tails than the normal distribution, or it contains some values that are influential for statistical inferences with the normal distribution, the multivariate t distribution becomes useful for multiple imputation as an alternative to the multivariate normal distribution. First, when the data have

longer tails than the normal distribution, the multiply imputed data sets using the t distribution allow more valid statistical inferences than those using the normal distribution with some "influential" observations deleted. Second, it is well known that the t distribution is widely used in applied statistics for robust statistical inferences. Therefore, when an incomplete data set contains some influential values or outliers, the t distribution allows for a robust multiple imputation method. Furthermore, the multiple imputation appears to be more useful than the asymptotic method of inference since the likelihood functions of the parameters of the t distribution given the observed data can have multiple modes. For a complete description of the MDA using the multivariate t distribution, see Liu (1995). See also Liu (1996) for extensions in two aspects, including covariates in the multivariate t models (as in Liu and Rubin, 1995), and replacing the multivariate t distribution with a more general class of distributions, that is, the class of normal/independent distributions (as in Lange and Sinsheimer, 1993). These extensions provide a flexible class of models for robust multivariate linear regression and multiple imputation. Liu (1996) described methods to implement the MDA for these models with fully observed predictor variables and possible missing values from outcome variables.

Estimation

214

10.4 Laplacian T-Approximation The Laplacian T-approximation (Sun et al., 1996) is a useful tool for Bayesian inferences for variance component models. Let p(0 I y) be the posterior pdf of 0 = (01i . . . , BP)T given data y, and let 77 = g(0) be the parameter of interest. Leonard et al. (1994) introduced a Laplacian T-approximation for the marginal posterior of ij of the form

x IT,I-112 p Y) A w/2 f (i 1w, 0n, T,) to be the marginal posterior pdf of q, where (e,11

p* (,qI Y)

11

T,n

=

U71

(w + p) Aq

TI

77

11

w+p-1'

U,? +w+p-1'

Qn

1 71

W

_ 1TQ-11

All

(10.7)

=

=

0**

alogp(01 Y) a0

0_0,

a2logp(0 I Y) a (99T)

=

0=0,,

0,1 +

w, 0*, T.n) denotes the pdf of q = g(0) when 0 possesses a multivariate t distribution with mean vector 0*, covariance matrix T,,, and degrees of freedom w. Here, 0,, represents some convenient and f (,q

I

approximation to the conditional posterior mean vector of 0, given q, and w should be taken to roughly approximate the degrees of freedom of a generalized multivariate T-approximation to the conditional distribution of 0 given q. When 0,, is the conditional posterior mode vector of 0, given rj, (10.7) reduces to the Laplacian approximation introduced by Leonard (1982)

and shown by Tierney and Kadane (1986) and Leonard et al. (1989) to possess saddlepoint accuracy as well as an excellent finite-sample accuracy, in many special cases. It was previously used for hierarchical models by Kass and Steffey (1989).

10.5 Sutradhar's Score Test

215

In the special case where q = aT B is a linear combination of the 0's, the approximation (10.7) is equivalent to p* (rl IY)

a

IT,

1-112

p

(0,71

Y)

),-(,, +p) /2tn (w,

aTo*, (aTT,a) -1)

where t, (w, p, T) denotes a generalized t pdf.

10.5 Sutradhar's Score Test Consider a random sample X1,. .. , Xn from a p-variate t distribution with the pdf

f (x) = 7

(v - 2)"/2IF ((v + p)/2) IRI1/2

7rp/2I' (v/2)

x

[v-2+(xj-µ)TR-1(xj-p)]

.

Note this is a slight reparameterization of the usual t pdf. The loglikelihood n

G = Elog f (xj) j=1

is a function of the parameters R, p, and v. Frequently in social sciences, and particularly in factor analysis, one of the main inference problems is to test the null hypothesis R = Ro when µ and v are known. Sutradhar (1993) proposed Neymann's (1959) score test for this test for large n. Le r = (rl1, ... , rhl, ... , rpp)T be the p(p+ 1)/2 x 1 vector formed by stacking the distinct elements of R, with rhi being the (h, l)th element of the p x p matrix R. Also let Ap(p+l)/2)

(A,,...,A

T

=

b (roe A, v)

and

(ro,N-,-0) 1

T

ro>

>v

where b(ro, µ, v), (ro, v), and q(ro, µ, are the score functions obtained under the null hypothesis r = ro, by replacing µ and v with their consistent estimates µ and v in b (ro

v)

=

ar '

(10.8)

Estimation

216

(ro

e

µ v)

=

aG aµ '

(10 . 9)

and

aG

(10 . 10)

av,

respectively. Furthermore, let Ti(ro, µ, v) = Ai - EP±i /3ij ryj, where /3i,j is the partial regression coefficient of Ai on y3. Then, Neyman's partial score test statistic is given by W (µ, v)

TT Mil

- (212M13)

MT12 M3 3

T

T,

M13 (10.11)

where T =_ [Ti (ro,

Tp(p+l)/2(ro ) µ, v)]T for i, r = 1, 2, 3; Mir

are obtained from Mir = E(-Dir) by replacing it and v with their consistent estimates; and Dir for i, r = 1,2,3 are the derivatives given by a2G D11

D12

D13 =

arar" a2G

araµ" a2G arav'

a2G

D22 = aµaft" a2G D23

aµav'

and

D33 =

02G av2

Under the null hypothesis r = ro, the test statistic W (A, v) has an approximate chi-squared distribution with degrees of freedom p(p + 1)/2. The test based on (10.11) is asymptotically locally most powerful. Clearly the implementation of this test requires consistent estimates of

10.5 Sutradhar's Score Test

217

µ, v as well as expressions for the score functions and the information matrix. The maximum likelihood estimates of µ and v are obtained by simultaneously solving n

tgj1Xj j=1

Eq.,-.l

j=1

and 71 (A, ro, = v) 0,

where qj = v - 2 + (Xj - µ)TRo(Xj - µ) and Ro is specified by the null hypothesis. The moment estimates of µ and v (which also turn out to be consistent) are n 1

Xj j=1

and

2 {2/32 - f (ro)J

02 - f (ro) where n

n(Xj-X)TRo(Xj-X)]

l2

j=1

is a consistent estimator of the multivariate measure of skewness (see, for example, Mardia, 1970b), and f (ro)

_ 3 E (roh)2 {rhh } 2 + h =1

12

{rhh hu h'

1

rohro h + (roh )2I ,

l

where rhh, and ro " denote the (h, h')th element of Ro and Ro 1, respectively.

10.5.1 Score Functions The score functions defined in (10.8), (10.9), and (10.10) are given by

b(r,Fi,V)

_ -2 InIp- (v+p)R-1Eq,-.1Aj I R-1, j=1

Estimation

218

n p)R-' E qj_1

(v +

(Xj - N-)

,

j=1

and

n{22 +log(v_2)+b()

ij (r, µ, v)

2b\2I]

-12 I(V+P) +Eloggj j=1

q,,

j=1

respectively, where denotes the digamma function and qj is a nonhomogeneous quadratic form given by qj = v - 2 + trR-1 B j with B j =

(X3 - tt)(Xj - µ)'.

10.5.2 Information Matrix By taking the second derivatives and then applying expectations, one can derive the elements of the information matrix. The first element takes the complicated form [m* (1,1), m* (1, 2), ... , m* (h, l), ... , m* (p, p)]

M11 =

,

where, for 1 > h, h, 1 = 1,. .. , p, m*(h, l) is the p(p + 1)/2-dimensional vector, formed by stacking the distinct elements of the p x p symmetric matrix n Mn,1

=

2

[rh

T

®(rl) J -

nv+ (2 p

R-1Qn,cR-1.

Here, r'k denotes the kth column of the R-1 matrix, and the (u, v)th element of the p x p matrix Qh,t is given by (v + 2) 2

P

P

+r r

VVrhZrtk (r r UV (v+4)2(v+p)(v+p+2)

Zu

v

+r

Zv

n- U )

'

i=1 j=1

where rms and rn,,s denote the (m, s)th element of R-1 and R, respectively. The second element of the information matrix M12 is zero. The third element M13 is formed by stacking the distinct elements of the symmetric matrix

n(p+2) (v - 2)(v + p)(v + p + 2)

R

1 .

10.6 Multivariate t Model

219

The remaining elements of the information matrix are given by

nv(v + p) (v - 2) (v +p+2) R

M22

M23 =

1

'

0,

and 1

M33

[i,(v+p\

4

, (l \21

v-4 2(v - 2)2

lnv(v2+vp-6p-2v-8)

+2 (v-2)2(v+p)(v+p+2) 10.6 Multivariate t Model Consider the following multivariate t model described in equation (9.2) of the preceding chapter

f (xi,...,X.)

-

F ((v + p)/2)

(irnv)n12 r (v/2)

IRIn12

n

x 1+ 1 V

t (xi - µ)T R-1 (xi - µ)

-(v+nv)12 .

(10.12)

i=1

In this section, we consider estimation issues associated with the corre-

lation matrix R and its trace tr(R).

10.6.1 Estimation of R Joarder and Ali (1997) developed estimators of R (when the mean vector µ is unknown) under the entropy loss function

L (u (A), R) = tr (R-1u (A)) - log IR-1u (A) I - p, where u(A) is any estimator of R based on the Wishart matrix A defined in (9.4). Based on the form of the likelihood function, the entropy loss function has been suggested in the literature by James and Stein (1961)

and is sometimes known as the Stein loss function. Some important features of the entropy loss function are that it is zero if the estimator u(A) equals the parameter It, positive when u(A) R, and invariant under translation as well as under a natural group of transformations of covariance matrices. Moreover, the loss function approaches infinity as

Estimation

220

the estimator approaches a singular matrix or when one or more elements (or one or more latent roots) of the estimator approaches infinity. This

means that gross underestimation is penalized just as heavily as gross overestimation. In estimating R by u(A), Joarder and Ali (1997) considered the risk function R(u(A), R) = E[L(u(A), R)]. An estimator u2(A) of R will be said to dominate another estimator ul (A) of R if, for all R belonging to the class of positive definite matrices, the inequality R(u2 (A), R) < R(ul (A), R) holds and the inequality R(u2 (A), R) < R(ul (A), R) holds for at least one R. Joarder and Ali (1997) obtained three estimators for It, by minimizing the risk function of the entropy loss function among three classes of estimators.

First, it is shown that the unbiased estimator ft = (v - 2)A/(vn) has the smallest risk among the class of estimators of the form cA, where c > 0, and the corresponding minimum risk is given by

R (R, R) = p log n - E E [log (xn+i_i) ] + p log 2

=1

\

-2pE (logy) , where r has the inverted gamma distribution given by (9.3). Second, the estimator R* = TD*TT, where T is a lower triangular matrix such that A = TTT and D* = diag(di,... , d p*) with dz defined by d*

_ -

v-2

Z

1

v n+p+1-2i'

has the smallest risk among the class of estimators TATT, where A belongs to the class of all positive definite diagonal matrices and the corresponding minimum risk function of R* is given by P

R (R*, R) =

P

log(n + 1 +p - 2i) +p log

v 2

E [log

(xn+1_

)]

2pE (log rr) ,

where T is as defined above. Furthermore, R* dominates the unbiased

estimator R = (v - 2)A/(vn).

10.6 Multivariate t Model

221

e Finally, consider the estimator R = SO(M)S, where A has the spectral decomposition A = SMST, with ¢(M) = D*M. Then the estimator R = SD* MST dominates the estimator R* = TD*TT.

10.6.2 Estimation of tr(R) Let 6 = tr(R) denote the trace of R. Joarder (1995) considered the estimation of 6 for the multivariate t model under a squared error loss function following Dey (1988). The usual estimator of 6 is given by 8 = cotr(A), where co is a known positive constant and A is the Wishart matrix defined in (9.4). The estimator 8 defines an unbiased estimator of 8 for co = (v - 2)/(vn) and a maximum likelihood estimator of 8 for co = 1/(n + 1) (see, for example, Fang and Anderson, 1990, page 208). Joarder and Singh (1997) proposed an improved estimator of 6 - based on a power transformation - given by

=

8

cotr (A) + coc {p

IAI1/r - tr (A) }

,

where co is a known positive constant and c is a constant chosen so that the mean square error (MSE) of 8 is minimized. Calculations show that

MSE (8)

= MSE (8)

+ c/31 + c2/92,

where 01

=

2co2E [(cotr (A) - 8) (p lAl1/n

- tr (A))]

(10.13)

and

02

= coE

[p jAj1/P

- tr (A)]

.

(10.14)

Thus MSE(8) is minimized at c = -/31/(2/92) and the minimum value is given by MSE(8) - /31 /(4/32). This proves that 6 is always better than the usual estimator in the sense of having a smaller MSE. The estimate of c is given by c = -/31/(2/92), where /91 and /32 are obtained by calculating the expectations in (10.13) and (10.14) using the numerous properties given in Section 9.1 and then replacing R by the usual estimator c0A.

It can be noted from Fang and Anderson (1990, page 208) that the estimators /91 and /32 are the maximum likelihood estimators of /31 and /32i respectively, provided R = c0A and co = 1/(n + 1). The following table taken from Joarder and Singh (1997) presents the percent relative efficiency of 8 over 8.

Estimation

222

R = diag(1,1,1)

R = diag(4, 2,1)

R = diag(25,1,1)

105.32 102.13

130.31

153.90

10

117.56

148.76

15

101.53

112.07

127.15

V

5

The numbers are from a Monte Carlo study carried out by generating 100 Wishart matrices from the multivariate t-model with n = 25 and p = 3.

10.7 Generalized Multivariate t Model Consider the generalized multivariate t model (9.7) discussed in the pre-

ceding chapter. The usual estimator of R is a multiple of the Wishart matrix of the form R = coA, where co > 0. Joarder and Ahmed (1998) proposed improved estimates for R as well as its trace and inverse under the quadratic loss function. The proposed estimators for R are

R = coA - c IAI1/P I,

(10.15)

where I is an identity matrix and c is chosen such that R is positive definite. For an estimator R* of R, let L(R*, R) = tr[(R* - R)2] denote the quadratic loss function and let R(R*, R) = EL(R*, R) denote the corresponding risk function. The relationship between R and R is rather involved. Defining the dominance of one estimator over another in the same manner as in Section 10.6.1, Joarder and Ahmed (1998) established that R dominates R for any c satisfying d < c < 0, where conp+2 p

d

FP((n-1)/2+1/p)

(10.16)

y FP ((n - 1)/2 + 2/p)

with co < pry/((n - 1)p + 2) or 0 < c < d, where d is given by (10.16) with co > pry/(np+ 2) and y by y = 72 /74 and y2 = E(rri), i = 1, 2, 3, 4. The risk functions of the two estimators are given by

R (R-, R) =

4p'Y4

- d Ir (R/P)

FP (Pl( 2/p) IRI2/P C 1\

1/P

Cc

+ {1 + (n - 1)coy4 (con - 2y)} tr (R2) +(n - 1)c02y4 (trR)2 and

R (R, R) =

{1 + (n - 1)coy4 (con - 2y)} tr (R2)

10.8 Simulation +(n - 1)c02ry4 (trR)2

223 ,

respectively. Now consider estimating the trace S = trR. The usual and

the proposed estimators are S = cotrA and 8 = cotrA - cp I A Il/p respectively, where co > 0 and c is such that the proposed estimator is positive. Joarder and Ahmed (1998) established that the corresponding risk functions are given by

R(8,6) = [(n-1)co{(n-1)coy4-2'y2}+1]82 +2(n - 1)coy4tr (R2) and

R (S, 8)

+ 2/p) IRI2iP = R (d S) + 4p2y4 Fp rp(n/2(n/2)

C

c - tr (R/p) d IRI1/P

respectively. It is evident that S dominates 8. Finally, consider estimating the inverse %F = R-1 with the usual and the proposed estimators = coA-1 and 'F = given by - co I A I-1/p I, respectively, where co > 0 and c is such that the _proposed estimator is positive definite. In this case, it turns out that 'F dominates ;P- for any c satisfying coA-1

d
- 'Y-2) Fp ((n - 1)/2 - 11p) co d = 4 ( n-2/p-p-2 y_4 l:p((n- 1)/2-2/p)

(10.17)

with co < (n - 2/p - p - 2)y_2/y_4 or 0 < c < d, where d is given by (10.17) with co > (n - 2/p - p - 2)y_2/y_4 and ryi = E(ri).

10.8 Simulation Simulation is a key element in modern statistical theory and applications. In this section, we describe three known approaches for simulating from multivariate t distributions. Undoubtedly, many other methods will be proposed and elaborated in the near future.

10.8.1 Vale and Maureli's Method Fleishman (1978) noted that the real-world distributions of (univariate) variables are typically characterized by their first four moments (that is, mean, variance, skewness, and kurtosis). He presented a procedure for generating nonnormal random numbers with these four moments specified. He accomplished this by taking a nonnormal variable X as a

Estimation

224

linear combination of the first three powers of a standard normal random variable Z

X = a+bZ+cZ2+dZ3.

(10.18)

To determine the constants, Fleishman expanded (10.18) to express the

first four moments of X in terms of the first 14 moments of Z. After considerable algebraic manipulation, Fleishman was able to represent the solution to the constants of (10.18) as a system of nonlinear equations. For a standard distribution (that is, with mean zero and variance one), the constants b, c, and d are found by simultaneously solving the following equations

b2+6bd+2c2+1542 = 2c (b2

+ 24bd + 105d2 + 2)

(10.19)

1,

=

-Y1,

(10.20)

and 24 {bd + c2 (1 + b2 + 28bd) + d2 (12 + 48bd + 141c2 + 22542) }

=

_Y2,

(10.21)

where ryl is the desired skewness and rye is the desired kurtosis. The constant a in (10.18) is determined by a

= -c.

(10.22)

Univariate nonnormal random numbers are then generated by drawing normal random numbers and transforming them using the constants a, b, c, and d in (10.18). Vale and Maureli (1983) extended Fleishman's procedure for multivariate nonnormal distributions with specified intercorrelations as well as specified moments. The procedure begins by specifying the constants necessary for Fleishman's procedure. For each variable independently, these are given by the solution of (10.19)-(10.22). Define two variables Zl and Z2 as from standard normal populations, and define the vector z as zT = [1, Z, Z2, Z31. The weight vector wT contains the power function weights a, b, c, and d: wT = [a, b, c, d]. The nonnormal variable X then becomes X = wT z. If rx1x2 denotes the correlation between two nonnormal variables X1 and X2 corresponding to the normal variables Z1 and Z2, it is then easily seen that rx1 x2 = wi Rw2, where X1 = w1 Z1, X2 = w2 z2 and R is the expected matrix product of z1

10.8 Simulation

225

and z2 given by 0

1

0

0

rzl z2

1

0

0 2rZ1 z2

3rz1 z2 0

0

3rzl z2

0

1

R =

+

1

64 z2 + 9rz, z2 J

Collecting the terms and using (10.22), a third-degree polynomial in rz1z2, the correlation between the normal variables Z1 and Z2, results rX1X2

= rz,z2 (bib2 + 3b1d2 + 3d1b2 + 9d1d2) + 2clc2rZlz2 +6d1d2rZ, z2.

Solving this polynomial for rz1 z2 provides the correlation required to obtain the desired post-transformation correlation rX1X2. These correlations can then be assembled into a matrix of intercorrelations, and this matrix can be decomposed to yield multivariate normal random numbers for input into Fleishman's transformation procedure.

10.8.2 Vaduva's Method Vaduva (1985) provided a general algorithm for generating from multivariate distributions and illustrated its applicability for multivariate normal, Dirichlet, and multivariate t distributions. Here, we present a specialized version of the algorithm for generating the p-variate t distribution with the joint pdf

f (x) =

r ((v +p)/2)

1/2 [1

+ I

(irv)p/ r (v/2) IR.I

XTR_1X-(v+11)/2

I

over some domain D in RP. It is as follows

(i) Initialize. (ii) Determine an interval I = [vo, vo] x ... x [vo, vt], where vo0 1

vo

= 0, =

v(p + 1)

V V-1 and v(p + 1)

1,

Estimation

226

(iii) Generate the random vector V* uniformly distributed over I. If RND is a uniform random number generator, then V* may be generated as follows

(a) Generate Uo, Ul, ... , UP uniformly distributed over [0,1] and stochastically independent. (b) Calculate Vi* = v° + (v2 - v°)Ui, i = 0,1, ... , p. (c) Take V* = (Vo , Vi, ... , VP).

(iv) If V* V D, then go to step (iii). (v) Otherwise, take V = V*. (vi) Calculate Y i = Vi/Vo, i = 1, ... , p. (vii) Take X = (Yl, ... , YP)T . Stop.

Note that the steps from (iii) to (v) constitute a rejection algorithm. The performance of this algorithm is characterized by the probability to accept V*. This probability can be calculated in the form 7rP/2 r

Pa

(v/2)

v

1

P

p + 2 2P(p + 1)r ((v + p)/2)IR11/2 C

which yields

lim pa

=

lim pa

= 0,

V-+00

0

and p-400

indicating inadequate behavior of the algorithm for large values of p and/or v.

10.8.3 Simulation Using BUGS A relatively simple way to generate a multivariate t involves a sampling

of z from gamma(v/2, v/2) and then sampling a multivariate normal y - NP(µ, R/z). This mode of generation reflects the scale mixture form of the multivariate t pdf. In BUGS the multivariate normal is parameterized by the precision matrix P; thus one programs a multivariate t pdf as follows to generate a sample of n cases (for Sigma[,], nu.2 and m u fl known) for (i in 1:n)

{z[i] " dgamma(nu.2, nu.2)

y[i, 1:q] ° dmnorm(mu, P.sc[,])}

10.8 Simulation

227

for (i in 1:q) {for (j in 1:q) {P[i, j] <- inverse(Sigma[,], i, j)

P.sc[i, j] <- z[i] * P[i, j]}} If one has observed multivariate data and wishes to assume multivariate t sampling, then in BUGS the dmt() form is available

for (i in 1:n) {y[i, 1:q] _ dmt(mu[], PC,], nu)} where nu is assumed known.

11

Regression Models

There is a large number of contributions (scattered in the literature and many of them motivated by economic applications) dealing with regression models with the error term distributed according to the multivariate t distribution. In this chapter, we shall discuss several of them.

11.1 Classical Linear Model Let the model for n observations Y = (yl, ... , y,,) ' be

Y = X/3 + E,

(11.1)

where X is an n x p design matrix with rank p, Q is a p x 1 vector of regression parameters with unknown values, and c is an n x 1 random error vector. For the usual t regression model it is assumed that the n elements of E have the multivariate t pdf f (E)

=

vvl2F ((n + v)/2) 7rn/2Qnr (v/2)

ET E1

-(n+v)/2

V + 92

In practice, there are several situations in which the model (11.1) is useful. Under (11.1), the least squares estimate of ,8 is ,Q

= (XTX) -1 XTy.

(11.2)

Zellner (1976) noted that this is also the maximum likelihood estimate of ,3. From Singh (1991), /3 is a minimum variance linear unbiased estimator and also a minimum variance unbiased estimator. The variancecovariance matrix for 0 is

Var H = E I Q - p) (

- Q)T 228

v a22 (XTX) -1

.

(11.3)

11.1 Classical Linear Model

229

Note that as v -4 oo, the above variance approaches (XTX)-1a2, which is the variance-covariance matrix in the normal case. Thus, for small and moderate values of v, the variances of the elements of A are inflated considerably, as compared to those for large values of v. Singh (1988) provided the following estimate of the degrees of freedom parameter 2 (2a - 3)

a-3

where (1/n) >

a =

( S (1/n)

=1

i

En

xT4 t (yi - XT Q) (Yi

t

and /3 is the least squares estimator given by (11.2). The maximum likelihood estimate of a2 is

(y - X/3)T (y n as in the normal case. For v > 2, a2

=

2

E (a) =

- X/3)

(n - p)au

n

where aU = vat/(v - 2) is the common variance of the elements of E. Thus, ET1 (n - p) is an unbiased estimator for au while

=

Q2

v((n -p)ETE

(11.4)

is an unbiased estimator for a2. In the class of estimators gETE, with q being a positive scalar, the minimal mean squared error estimator for a2 is (with v > 4) a2

v-4

=

v(n -p+ 2)

-T-

E E,

(11.5)

while the minimal mean squared error estimator for a,21 in this class is (v - 4)eTE/{(v - 2) (n - p + 2)}. The variances of the unbiased and the minimal mean squared error estimators of a2 are

Var (a2)

=

2a4 n - p + v - 2

n-p

v-4

(11.6)

Regression Models

230 and

Var (v2)

(v-4)(n-p+v-2) 4

= 2(n - p) (v

- 2)2(n - p + 2)2

(11.7)

a'

respectively. Since a2 is an unbiased estimator for a2, the variance (11.3) can be unbiasedly estimated by

_ (XTX)-1 E- E I.

Var (Q)

n-p

Similarly, (11.4) and (11.5) can be estimated by a

_T_

(2a-3)(n-p)' E

(11.8)

3 T^ (2a-3)(n-p+2)' E

(11.9)

and a2

respectively. The estimates for the variances given by (11.6)-(11.7) are

Var (a2)

=

2(a*2)2n-p+v-2 n-p

v-4

and Var (Q2)

-

- 4)(n - p + v - 2) 2( n- p)(v(v-2)2(n-p+2)2 (a

2

*2

)

respectively, where a*2 may be taken as a2 given in (11.8) or as 72 given in (11.9).

It is important to note that, even though the elements of E have the nonnormal pdf and are not independent, tests and intervals based_ on the usual t and F statistics still remain valid. For example, t = (/3 /3z)/{s mii}, where mii is the (i,i)th element of (XTX)-1 and s2 =

E E/(n - p), has the usual Student's t distribution, and thus probability -T statements based on this statistic will be appropriate. Also, s2/a2 has the usual F distribution with degrees of freedom v and n - k. This fact can be used to construct confidence intervals for and test hypotheses about a2. Singh et al. (1995) proposed the generalized estimator ,Q9 = g(t),3

for Q, where t = ETE//3T XT X/3 has at least the first k > 6 moments finite and g(t), satisfying the validity conditions of Taylor's series expansion and having the first three derivatives with respect to t bounded,

is a bounded function of t such that g(O) = 1 and g(t) = 0(1) as

11.1 Classical Linear Model

0=

231

6T XT X,6 _+ co.

It should be noted that the maximum likelihood estimator /3 and the estimators considered by Singh (1991) are all particular cases of /39. Singh et al. (1995) investigated the bias p(%39) = E[(l39 - p)TQ(p9 - p)] of the generalized estimator when Q is a positive definite matrix. It was established that E

=Q+ \/3 ^9)

(n - p)va2g'(0) I 1 - vo2(p - 2) 0(v - 4) 0(v - 2) IL

+vol2(n - p+2)g"(0)1Q+O 20(v - 4)g (0)

v92

(h.)

I

(n - p)7 2a4g'(0)

v-2 tr { (XTX)_' Q} + (v-2)(v-4)0 x [2tr { (XTX)

-1

Q} + (n - p + 2)g'(0) - 4 /TQA B

Since p(/39) < p(/3), one observes that N9 is dominant over the maximum likelihood estimator ,Q. Also in the class /39, there exists better estimators than those considered in Singh (1991). Sutradhar (1988b) considered testing Ho : C6 = 0 versus H1 : C/3 # 0, using the classical F statistic

W =

W2 - W1 W1

where

Wl =

v-2 2

In-X(XTX)-1XT)Y

vQ(

J

is the residual sum of squares of the full model (11.1) and W2

=

y-2 (I,,-Z(ZTZ)-1ZT)Y V012

is the residual sum of squares of the reduced model

E (Y)

= Za,

(11.10)

which is obtained from (11.1) by using the restriction under Ho. In Ho,

C is an r x p matrix of known coefficients with rank(C) = q, and, in the reduced model (11.10), Z denotes the new design matrix of order

Regression Models

232

n x (p - q) and a is a vector of (p - q) parameters. Sutradhar (1988b) established that the pdf of W is given by 2

.f (w)

=

°O

v-2

Q6/(,.-2) (k + 1,

r

v

k+

q n-p 22

k=o

where

F(a + I'(a)F(b)(1 + x)a+b b)xa-1

/3X (a, b)

=

and 8 is the noncentrality parameter given by

6=

v -22 (iT vo'

F

(XTX

- XTZ (ZT Z)-1 ZT X) ,Q.

Sutradhar also computed the corresponding power of the test, yielding the expression 00 / p E '861(v-2) (k + 1, 2 1) Imo (n 2 , k+ 2 ) 2 v 2

k=O

where uo = 1/[1 + (q/(n - p))Fq,n_p,a] and Iu(a, b) denotes the incom-

plete beta function ratio. As v -> no, this expression reduces to the power of the F test under normality (Tiku, 1967). The distribution of future responses given a set of data from an informative experiment is known as a predictive distribution. Haq and Khan (1990) derived the predictive distribution for (11.1). Rewrite (11.1) in the equivalent form y = 8X + ae and let Yf be a future response corresponding to the design vector x f, that is, yf = /3xf + ae f. Haq and Khan (1990) showed that the predictive pdf of Yf is given by f (Yf oc

I Y)

(n-p+1)/2

[i + s-2(Y) {yf - b(Y)xf} (1 - xfTA-'xf) {yf - b(Y)xf IT J

where b(e) = eXT(XXT)-1, s2(e) = (e - ;&) (e - e) T, e = b(e)X, and A = XXT + x fxf . Thus, for the given informative data y, the predictive distribution of Yf is t with mean vector b(y)x f, variance-covariance matrix (n - p)s2(y)/{(n - p - 2)(1 - XT A-1x f)}, and degrees of freef dom n - p. A prediction interval of the desired coverage probability can easily be obtained by using the standard t-table. Note that the predictive distribution does not depend on the degrees of freedom parameter of the original t distribution. For a set of n' future responses given by

11.2 Bayesian Linear Models

233

Yf = )3X f + ore f, Haq and Khan (1990) noted similarly that the predictive distribution of y f is n'-variate t with mean vector b(y)Xf, variancecovariance matrix I I,,,, - XfQ-1X f-1/2 s(y) (where Q = XXT + Xf X f ), and degrees of freedom n - p. It is to be noted that the distri-

bution of (n-p)s-2(y)(Yf -b(Y)Xf)(In,' -XfQ-1Xf)(Yf -b(Y)Xf)T

is F with degrees of freedom n', and n - p. This distribution can be utilized for determining the prediction region for a set of future responses with any desired coverage probability.

11.2 Bayesian Linear Models In his classical paper, Zellner (1976) provided a Bayesian analysis of the linear model (11.1). Consider the diffusion prior for /3 and Q2, that is,

p(/i,a2)

oc

(11.11)

T2 ,

where 0 < o, 2 < oo and /3a E R, i = I,-, k. Then, assuming that v is known, the posterior pdf of the parameters is

a { A (N) } -n/2

P (Q, Q 2 I Y, V)

[vat /A (Q)] A (i3) [1 + vat /A

v/2-1 (,8)](,,+, )/2 1

where A(,Q) = (y - X)9)T (y - X,3). It follows that the conditional posterior pdf of ,Q given a2 and v is in the form of a multivariate t pdf with mean /j (the least squares estimate in (11.2)). The corresponding conditional posterior covariance matrix is given by vo2 + (n - p)s2

T

v+n - P - 2 (X X)

-1

provided that n - p + v > 2, where (n -p)82 = (y - X/3)T (y - X/l). As v -+ oo, the conditional posterior pdf for ,Q and a2 approaches a mul(XTX)-1a2, tivariate normal pdf with mean ,Q and covariance matrix which is the usual result for the normal regression model with the diffuse prior pdf (11.11). The marginal posterior pdf for ,Q is _ yn'2, ( {(n-p)s2+(/i-,Q)7

P(/jJY,v)

oc

XTX (13 -)

(11.12)

which is in the form of a p-dimensional t pdf and does not depend on the value of v. In fact, (11.12) is precisely the result that one obtains in the Bayesian analysis of the normal regression model with the diffuse

Regression Models

234

prior for the parameters shown in (11.11). The marginal posterior pdf for a2 is

all (v/2)-1 p

(0,21

y, v)

\ s2 /

-(v+n-p)/2

vO,2

(\1 +

OC

(n - p)s2 )

from which it follows that a2/s2 has the F pdf with degrees of freedom v and n - p, a result paralleling the classical results mentioned in the preceding section. From properties of the F distribution, the modal

value of o, 2/s2 is ((n - p)/v)((v - 2)/(n-p+2)), when v > 2 and its mean is (n - p) / (n - p - 2) when n - p > 2. Also, as v -4 oo, the posterior distribution of vs2/v2 approaches a chi-squared distribution with degrees of freedom n - p, a distributional result that holds for the Bayesian analysis of the usual normal regression model with diffuse prior assumptions. Finally, note that the posterior pdf for no'2 / (y - XA)T (y X,3) is F,,,n. The natural conjugate prior distribution for o, 2 and ,8 is the product of the marginal F pdf for a2 times a conditional p-dimensional t pdf for ,3 given Q2, that is, P (Q, Or2I ')

= PF (0,21 _) Ps (QI a2, ')

,

where 2

PF

(01 2

2

2) (v-2)/2

(Vol /vasa)

say vas v)

oc vQ2/vaSa)(v+va)12

(1 +

(where va > 0, Sa > 0, and 0 < a < oo) and

PS (QI

2

Q, A, I/a)

oc

Oa P f a +

1

Q2 a

(-

(2v+P)l2

)T

A (Q

where /32 E R, i = 1, ... , p, A is symmetric and positive definite, fi is

the prior mean vector, 1/a = v + va, and va = (vasa + vat)/Va. As with the natural conjugate for the usual normal regression model, it is seen that A and u2 are not independent in the natural conjugate prior distribution in (11.13). If the natural conjugate prior distribution is thought to represent the available prior information adequately, it can be used for obtaining the posterior distribution; see the appendix in Zellner (1976) for details.

11.3 Indexed Linear Models

235

11.3 Indexed Linear Models Lange et al. (1989) and Fernandez and Steel (1999) provided a farreaching extension of (11.1) to handle the situation when yj's are assumed to have the t distribution with degrees of freedom vi and parameters 1z = gi (O) and R = hi (p) indexed by some unknown parameters 0 and 0. Lange et al. (1989) suggested an EM algorithm for estimation. They also considered methods for computing standard errors, developed graphical diagnostic checks, and provided applications to a variety of problems. The problems include linear and nonlinear regression, robust estimation of the mean and covariance matrix with missing data, unbalanced multivariate repeated-measures data, multivariate modeling of pedigree data, and multivariate nonlinear regression. They also derived the expected information matrix for (0, 0, v) for one observation in the form

_

E 8 log L 1

V+ P avT

v+p+2 aoi

C99 99

a log L

l aoiaoj/

_

V+ P

2(v+p+2)

tr R-1

(

2 1 2 (v+p+)

E

alogL)

__

tr

81ogL

1

1

2

v+p

2( 2) 1

1

R-1

1' (vl 2

OR1

a0i p

\2/ + v(v+p)

v+2

v+p+v(v+p+2)

where

a0,'

R-1 aR tr R-1 s:),

(v+p)(v+p+2)tr

_

av

9R R_1 aR a0; a0i

00i0v

E (avav)

R-1

'

0 (x) = d2 log F(x)/d2x is the trigamma function. The remaining

elements of the matrix are zero.

In an important paper, Fernandez and Steel (1999) revealed some pitfalls with a model of the above kind. Under a commonly used noninformative prior, they showed that Bayesian inference is precluded for certain samples, even though there exists a well-defined conditional distribution of the parameters given the observables. They also noted that global maximization of the likelihood function is a vacuous exercise since

236

Regression Models

the latter becomes unbounded as one tends to the boundary of the parameter space. More specifically, let 1(9, a, R, v) be the likelihood function for n independent observations yj assumed to have the t distribution with mean vector gi(9), common covariance matrix Q2R, and common degrees of freedom v. For given values of 0 = Bo, R = Ro, and v = vo, let 0 < s(9o) < n denote the number of observations for which yi = gi(9o). Then the following hold

(a) If vo

<

Ps (90) n-s(9o),

then

=

lim l (9o, o,, Ro, vo)

a-0

co.

(b) If Ps (eo)

v0

n-s(9o)'

then

liml(90,o,,Ro,vo)

E

(0,o0).

(c) If v

>

ps(90)

then

l' ol(9o,o,,Ro,vo) =

0.

It is evident from this result that one can determine a value 9o such that yi = gi(9o) holds for at least one observation and the likelihood function does not possess a global maximum. Indeed, for sufficiently small values of v, one can make 1(90, o,, R0i vo) arbitrarily large by letting a tend to zero. These pitfalls arise as a consequence of the (sometimes neglected)

fact that the recorded data have zero probability under the assumed model. Fernandez and Steel (1999) proposed and illustrated a Bayesian analysis on the basis of set of observations that takes into account the precision with which the data were originally recorded.

11.4 General Linear Model

237

11.4 General Linear Model Rubin (1983) and Sutradhar and Ali (1986) considered the general linear model set up in the form

Y = OX +'E'

(11.14)

where X is a k x n design matrix with rank k, /3 is a p x k matrix of regression parameters with unknown values, and E is a p x n random error matrix. It is assumed that the error variables Eij satisfy =

E(Eij)

di,j=1,...,p,

0,

a2 Aii,

E(Ea) E (eij elj)

di,j = 1,...,p,

= a2Ail, Vi, j, l = 1, ... , p,

and

=

E (Eijelj,)

0,

V2, 1, j :Aj

where Aij are unknown parameters. Furthermore, it is assumed that, for a given a, the errors El, ... , En, are independently and normally distributed, with the distribution of Ej = ( 6 1 j ,.. . , Epj)T being N(0, a2A) f o r j = 1, ... , n while a is assumed to be a random variable having an inverted gamma distribution with the pdf given by 2 (v/2)(v+1)12

f (a)

=

(v/2)

VI'v

a -(v+1) exp

v

1-2a 2

,

where v is an unknown parameter. Under these assumptions, one can show that the joint distribution of error variables is T T f (E1 , ... , En)

_

(v - 2) vl2r ((v + np)/2) IRIn/2

7rnp/2r (v/2) n

x

-(v+np)/2

j

j-1

where R = vA/(v - 2). It then follows that E(Ej) = 0, E(e2c ) = R and E(EjEB) = 0 for j 0 s, j, s = 1, ... , n. Sutradhar and Ali (1986) provided a least squares estimator for /3 as well as moment estimators for R and P. The least squares estimator is

=

(XXT)-1 XYT

Regression Models

238

while the moment estimators are given by n

1

R =

n

j=1

(yj

T

q

- NXj) (Yj -'

xj)

and

2 36 i=1

i

Evii j

)/(

3E

i

i=1

nI:E i

j

where Eij are the so-called estimated residuals expressed as the difference k

Yij - 1:9irXrj.

cij

r=1

All three estimators Q, R, and v are consistent as n -4 oo. Let Y = (yi i ... , yn)T , where yj = (yl j, ... , ypj )T . Let Y* denote the stacked random vector corresponding to Y, so that Y* = (y11) , yP1, y12, ..., yP21 ..., y1n, ..., yPn )T Let ,3* and e* be the corresponding stacked random vectors. Then the model (11.14) can be written in terms of Kroneckor products as

Y* = (I® ® XT) /3* + e*.

(11.15)

Suppose one wishes to test the hypothesis that Ho : 0* = Bo versus H1 : 9* $ 0*. In the case where v and R are known, Sutradhar and Ali (1986) showed that a suitable test statistic is D

v

v

2

(8*

- Bo}T {R 0

(XXT)-1

0*)

Lower values of this statistic D will favor Ho while higher values. will direct the rejection of Ho. Actually, it can be shown that the pdf of D is Vv12dkP/2-1 00

f(d) =

r ((v + kp)/2 + 2j)

r (kp/2 + j) F(j + 1)

I (v/2)

x (Ad)3 (A + v + d)

(v } kP)/2-2j

where

A=

v v

2

(9* - 9o)T o-1 (B*-6*) o .

Note that, under Ho : 0* = 0 , D/(kp) has the usual F distribution with degrees of freedom kp and v, whereas the analogous test for the classical

11.5 Nonlinear Models

239

MANOVA model has the chi-squared distribution with degrees of freedom kp. Also note that the power of the test changes under H1, whereas the similar statistic has the noncentral chi-squared distribution for the usual normal model. In the case where v and R are not known, since v and ft are consistent estimators, an F test based on D = vUT U/(v -2),

f =0

(9 - 60), may still be approximately valid. Little (1988) extended the general linear model (11.14) to handle incomplete data. The methods for estimation employed are maximum likelihood (ML) for multivariate t and contaminated normal models. ML estimation was achieved by means of the EM algorithm and involves minor modifications to the EM algorithm for multivariate normal data. where

11.5 Nonlinear Models Nonlinear models involving multivariate t distributed errors have been studied relatively recently. Chib et al. (1991) considered nonlinear regression models with errors that follow the multivariate t distribution with degrees of freedom v. For an n x 1 vector of observations y, the model is specified by

y = h (X, /3) + e,

(11.16)

where X is an n x r matrix of regressors, /8 is the regression coefficient vector, is a vector function of and e is the error vector. It is assumed that e I X, /3, rl, T, v has an n-variate t distribution with zero mean vector, covariance matrix (1/T)V(X,27), and degrees of freedom

v. On can see that (11.16) reduces to (11.1) simply by setting r = p, h(X, p) = X/3, and V (X, ,q) = In,. The sampling density resulting from (11.16) is a t pdf, which can be represented as the following scale mixture of normal pdfs 00

f (YIX,w) = ff(YIXz,w)f(zIX,w)dz, where f (y I X, z, w) is an n-variate normal pdf with mean vector and covariance matrix 1/ (zr)V (X,71) and f (z I X, w) is a gamma pdf with parameters (v/2, v/2). Note that the proper pdf, f (z I X, w), is independent of X and does not involve parameters other than v. In the classical linear model due to Zellner (1976), the marginal posterior of the regression parameter, Q, is unaffected by the multivariate

t assumption (see Section 11.2). This result was extended by Chib et

240

Regression Models

al. (1998), Osiewalski (1991), and Osiewalski and Steel (1990) for elliptically distributed errors. For the nonlinear model above, Chib et al. (1991) provided the following sufficient conditions under which the posterior of v, p(v I y, X), coincides with the prior, p(v)

For proper priors p(w), if v is independent of ((3, ,q, Tz), then v is independent of (y, X). For improper priors of the form p(w) = p(T)p(,8, rj)p(v), where p(T) x 1/7-, T > 0 and p(v) is proper and functionally independent of (T, /3, rj), if the posterior of v exists, then p(v I y, X) = p(v).

12 Applications

Due to limitations on the size of this book and since the aim is to collect and organize results on multivariate t distributions, in this short chapter we collect and present a small number of relatively recent applications

of multivariate t distributions. The treatment is by no means exhaustive. Some other applications - in particular those related to Bayesian inference - are mentioned in the previous chapters (see Chapters 1, 3, 5, 10, and 11).

12.1 Projection Pursuit Exploratory projection pursuit is a technique for finding "interesting" low p-dimensional projections of high P-dimensional multivariate data; see Jones and Sibson (1987) for an introduction. Typically, projection pursuit uses a projection index, a functional computed on a projected density (or data set), to measure the "interestingness" of the current projection and then uses a numerical optimizer to move the projection direction to a more interesting position. Loosely speaking, a robust projection index is one that prefers projections involving true clusters over those consisting of a cluster and an outlier. A good robust projection index should perform well even when specific assumptions required for "normal operation" fail to hold or hold only approximately. In a paper that was awarded the Royal Statistical Society Bronze Medal, Nason (2001) developed five new indices based on measuring divergence from the multivariate t distribution with the joint pdf f (x)

-

I' ((v +p)/2) 7fp/2 (v - 2)p/2r' (v/2) (1 241

xTx

v - 2)

- (-+p)/2

Applications

242

that are intended to be especially robust. The first three indices are all weighted versions of the L2-divergences from f for v > 3. They are given by

ITa

2

f{g(x)_f(x)}2f(x)dx

=

for a = 0, 1/2, 1. Nason (2000) derived an explicit formula for the case a = 0. The fourth index is the Student's t index defined by

ITI

fgl_2/(V+P)(x)dx.

V

This index is minimized over all spherical densities by f (x). Specifically, it satisfies the inequality

(v+p-2)F((v+p)/2)

>

-

IV

7rp/2(v - 2) 1+p/211 (v/2)

for all spherical densities g with equality if and only if g = f almost everywhere. The proof of this result uses the fact that the index can be represented as the sum of two F-divergences (Vajda, 1989). Through both numerical calculation and explicit analytical formulas, Nason (2001)

found the the Student's t indices are generally more robust and that indices based on L2-divergences are also the most robust in their class. A detailed analytical exploration of one of the indices (ITo 2) showed that it acts robustly when outliers diverge from a main cluster but behaves like a standard projection index when two clusters diverge, that is, its behavior automatically changes depending on the degree of outlier contamination. The degree of sensitivity to outliers can be reduced by increasing the degrees of freedom v of the ITo2 index to make it behave increasingly like Hall's index (Hall, 1989) as v -4 oo.

Using the transformation x = tan(g), Nason further developed the orthogonal expansion index given by 2

ITL2

=

V

x

/2

2

f ir/2 l go (0) - x{

2

cos4 9 } dB, JJJ

where go is the pdf of the transformed projected data X. Using the Fourier series expansion of go(9) on [-it/2,7r/2], go (0)

=2+

{ancos(2n0) + an sin(2n8)} n=1

,

12.2 Portfolio Optimization

243

where

an =

2

it/2

g(O) cos(2n9)dO -7r/2

and 2

lr/2

IT

-7r/2

=

bn

g(O) sin(2n9)dO,

the index ITL2 can be expanded as TL2

13,1/2

_

7 2

-

2

3 12 (ao

2

c

-

27

I

+ (al

7r) + (a2

2

47r

)

00

+Ea2n+:L bn n=3

n=3

12.2 Portfolio Optimization There are a number of places in finance where robust estimation has been used. For example, when a stock's returns are regressed on the market returns, the slope coefficient, called beta, is a measure of the relative riskiness of the stock in comparison to the market. Quite often, this regression will be performed using robust procedures. However, there appear to be fewer applications of robust estimation in the area of portfolio optimization. In the problem of finding a risk-minimizing portfolio subject to linear constraints, the classical approach assumes normality without exceptions. Lauprete et al. (2002) addressed the

problem when the return data are generated by a multivariate distribution that is elliptically symmetric but not necessarily normal. They showed that when the returns have marginal heavy tails and multivariate tail-dependence, portfolios will also have heavy tails, and the classical procedures will be susceptible to outliers. They showed theoretically, and on simulated data, that robust alternatives have lower risks. In particular, they showed that when returns have a multivariate t distribution with degrees of freedom less than 6, the least absolute deviation (LAD)

estimator has an asymptotically lower risk than the one based on the classical approach. The proposed methodology is applicable when heavy tails and tail-dependence in financial markets are documented especially at high sampling frequencies.

244

Applications

12.3 Discriminant and Cluster Analysis In the past, there have been many attempts to modify existing methods of discriminant and cluster analyses to provide robust procedures. Some of these have been of a rather ad hoc nature. Recently the multivariate t distribution has been employed for robust estimation. Suppose, for simplicity, that one utilizes two samples in order to assign a new observation into one of two groups, and consider the joint distribution

f xl,x2) vl-v --2r (v + np/2) ,np/2

IRIn/2

2

- (v+np)/2

n;

x 1(v - 2) +

(xij i_1 j=1

- µi)TR 1(x2 -µ2)

(12.1)

of the two samples Xi = (X11, ... , X1n1) and X2 = (X21, ... , X2n2) of sizes nl and n2, respectively. In (12.1), n = ni + n2. The (nl + n2)pdimensional t distribution (12.1) was proposed by Sutradhar (1990). It is evident that the marginals are distributed according to f (x2a)

-

v _-217 (v + p/2) 7rp/2 IRIn/2

(v+p)/2

x [(v -

2) + (xij - µi)T R-1 (xij - µi),

(12.2)

which is a slight reparameterization of the usual multivariate t pdf. Let 7r1 and 7r2 denote the two t-populations of the form (12.2) with parameters (µl, R, v) and (µ2, R, v), respectively. Fisher's optimal discrimination criterion is robust against departure from normality (Sutradhar, 1990), and it assigns the new observation with measurement X to 7r1 if d(x)

1

_ (Al - µ2)T R-lx - 1 (Al - µ2)T R-1 (µl + µ2) ? 0;

otherwise, it assigns the observation to 7r2. But even though the classification is based on the robust criterion, the probability of misclassification depends on the degrees of freedom of the t distribution. If el and e2 are probabilities of misclassification of an individual observation from 7r1 into 72 and from 72 into 7rl, respectively, then ei

=

v --2 r (v + 1/2)

-°/2 00

{(v - 2) +

z2}-(v+1)/2

dz

12.4 Multiple Decision Problems

245

for i = 1, 2, where A2 = (Al - µ2)T R-1 (µl - µ2). Calculations of el and e2 for selected values of 0 and v (Sutradhar, 1990) suggest that if a sample actually comes from a t-population (12.2) with degrees of freedom v, then the evaluation of the classification error rates by normal-based probabilities would unnecessarily make an experimenter more suspicious. Sutradhar (1990) illustrated the use of the preceding discrimination approach by fitting the t distribution to some bivariate data on two species of flea beetles. McLachlan and Peel (1998), McLachlan et al. (1999), and Peel and McLachlan (2000) used a mixture model of t distributions for a robust method of mixture estimation of clustering. They illustrated its usefulness by a cluster analysis of a simulated data set with added background noise and of an actual data set. For other recent methods for making cluster algorithms robust, see Smith et al. (1993), Dave and Krishnapuram (1995), Jolion et al. (1995), Frigui and Krishnapuram (1996), Kharin (1996), Rousseeuw et al. (1996), and Zhuang et al. (1996).

12.4 Multiple Decision Problems The multivariate t distribution arises quite naturally in multiple decision problems. In fact, it is one of the earliest applications of this distribution in statistical inference. Suppose there are q dependent variates with means 01 i ..., 9h, ..., 9q, respectively, and that one has estimators 9h of Oh, h = 1, . . . , q available, which are jointly distributed according to a q-variate normal distribution with mean 0h, h = 1, ... , q, and covariance matrix o-2R, where R is a q x q positive definite matrix and 0.2 is an unknown scale parameter. Let s2 be an unbiased estimator of Q2 such that s2 is independent of the 0h's and vs2/Q2 has the chi-squared distribution with degrees of freedom v. Consider p < q linearly independent linear combinations of Oh's, q

mi =

CihOh = cT B, h=1

for i = 1,.. , p, where ci =

(Ci1, ... , Cih, ... , Ciq)T is a q x 1 vector of known constants. The unbiased estimators of the mi's are .

q

mi =

Ciheh = CT B, h=1

Applications

246

each of which is a normally distributed random variable with mean mi and variance cTRci. Then Yi

=

mi - mi s ciT Rci

i = 1,...,p

is a Student's t-variate and Y1, ... , Y, have the usual p-variate t distribution with degrees of freedom v, zero means, and the correlation matrix {8iv} given by cTRc,,,

cTRcic,T.Rcu

For multiple comparisons, one computes the one- and two-sided confidence interval estimates of mi (i = 1, . . . , p) simultaneously with a joint confidence coefficient 1 - a, say. These estimates are given by (Dunnett, 1955)

ii ± h1s cTRci and

i12i ± has

cTRci,

respectively, where the constants h1 and h2 are determined so that the intervals in each case have a joint coverage probability of 1 - a. The constants h1 and h2 can be computed using the methods discussed in Chapter 8.

12.5 Other Applications Bayesian prediction approaches using the multivariate t distribution have attracted wide-ranging applications in the last several decades, and many sources are available in periodic and monographic literature. Chien (2002) discusses applications in speech recognition and online environmental learning. In experiments of hands-free car speech recogni-

tion of connected Chinese digits, it was shown that the proposed approach is significantly better than conventional approaches. Blattberg and Gonedes (1974) were one of the first to discuss applications to security returns data. For other applications, we refer the reader to the numerous modern books on multivariate analysis and to the Proceedings of the Valencia International Meetings.

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Index

equi, 25, 34, 127, 134, 135, 143, 172, 175, 178, 182, 183, 187

F matrix, 20 F-divergence, 242 L2-divergence, 242

intraclass, 112 quasi-decomposable, 189 singular, 189 cumulant generating function, 121, 122,

t-ratio distribution, 60

Fortran, 153, 154 acceptance-rejection sampling, 157, 158 Appell's hypergeometric function, 32, 33 Appell's polynomials, 10 association, 20

Bartlett's decomposition, 213 Bayesian estimator, 159-161 Bayesian inference, 2, 50, 112, 213, 214, 233-236, 241, 246 Behren-Fisher density, 32, 33 Bessel function first kind, 42 second kind, 42 beta function, 78 Bonferroni's inequalities, 177 BUGS, 226, 227 characteristic function, 36-41, 43-45, 48, 76, 120, 135, 200 Cholesky decomposition, 151, 156 cluster analysis, 52, 212, 244, 245 complementary error function, 143, 144 complete elliptical integral first kind, 48, 78, 79 second kind, 48, 78 concomitant order statistics, 207, 208 cone, 145, 146 confluent hypergeometric function, 47,

205

decision theory, 12 dependence coefficient, 25 digamma function, 22, 23, 204, 218 discriminant analysis, 244 distribution asymmetric multivariate t, 97, 99 bivariate t, 13, 15, 21, 56, 72, 74, 80, 81, 106, 127, 148, 161, 170, 174, 181, 183, 207 bivariate Cauchy, 68, 80 bivariate chi-squared, 71, 73 bivariate normal, 15, 20, 53, 63, 66, 68, 69, 71, 73-75, 80, 119, 129, 148, 154, 170, 174, 209 bivariate Pearson type VII, 80 central bivariate t, 72, 127 central matrix-variate t, 113-116 central multivariate t, 1, 16, 17, 19, 22, 29, 87, 88, 90, 99, 114-117, 126, 131, 135, 140, 165, 170, 172, 175, 182, 204, 205 central univariate F, 20 complex matrix-variate t, 120 complex multivariate t, 119, 120 complex multivariate normal, 119, 120

complex univariate normal, 120 compound normal, 96 conditionally specified bivariate t, 76 Dirichlet, 84, 225 doubly noncentral bivariate t, 71 doubly noncentral multivariate t, 93

71, 90 consistency, 9 convex polyhedra, 148, 150 correlation structure

decomposable, 171, 189, 190 269

270

Index

doubly noncentral univariate t, 71, 93 doubly noncentral univariate chi-squared, 93 Fisher's z, 34 gamma, 226, 239 generalized t, 94, 96, 97, 101 Gumbel, 106 infinite matrix-variate t, 119 inverse gamma, 94 inverted Dirichlet, 126 inverted gamma, 192, 196, 220, 237 matrix-variate t, 97, 112-115, 118, 119

matrix-variate beta, 117 matrix-variate inverse Wishart, 118 matrix-variate normal, 118 mixture of normal, 9 mixture of two normal, 200 multivariate F, 100, 110 multivariate t, 1, 8, 16, 22, 24, 30, 36, 38, 41, 44, 50, 87, 88, 94, 98, 102, 103, 106, 113, 115, 117, 120, 126, 127, 135, 140, 145, 151, 153, 154, 159, 161, 165, 170, 172-174, 183, 184, 187, 188, 191, 204, 210, 213-215, 223, 225-228, 233-236, 239, 241, 243-246

multivariate Bessel, 9 multivariate Cauchy, 1, 9, 22, 25, 40, 41, 100, 192 multivariate elliptical, 2, 94, 96, 100, 120, 121, 123, 125, 126, 196, 200, 201, 240, 243 multivariate logistic, 9

multivariate normal, 1, 2,

9, 12, 13, 15, 18, 24, 31, 50, 52, 73, 87, 90, 93-96, 98-100, 103, 112, 119, 123, 125, 140, 152, 155, 156, 165, 171, 181, 192, 196, 205, 213, 225, 226, 233, 239, 245

multivariate Pearson type II, 9 multivariate Pearson type VII, 2, 9, 40, 41

multivariate uniform, 157, 158, 226 multivariate Wishart, 7 noncentral bivariate t, 63, 66, 68, 69, 73

noncentral complex Wishart, 197 noncentral matrix-variate t, 113 noncentral multivariate t, 1, 23, 87, 90, 93, 139, 140

noncentral multivariate Cauchy, 23 noncentral univariate F, 19, 34 noncentral univariate t, 63 noncentral univariate chi-squared, 71, 209, 239

noncentral Wishart, 197

power exponential, 9 scale mixture of normal, 194, 239 skewed multivariate t, 98, 100, 102-105, 107, 109, 112 skewed multivariate Cauchy, 100 skewed multivariate elliptical, 101 skewed multivariate normal, 98, 103 skewed univariate t, 83, 84, 106, 107, 109

spherical, 9, 96 standard normal, 49, 57, 63, 80, 82, 98, 130, 132, 133, 136, 144, 166, 175, 177, 181-184, 186, 224 stepwise multivariate t, 90, 91 Student's t, 1, 13, 23, 28, 30, 36, 40, 41, 49, 52, 53, 56, 63, 64, 66, 67, 75, 79, 80, 82, 84, 92, 98, 99, 101, 105, 111, 116, 129, 137, 145, 150, 156, 161, 163, 167, 169, 170, 175, 179, 185, 230, 246

symmetric multivariate t, 111 trivariate t, 21, 35, 183 trivariate normal, 67, 121, 123 univariate F, 107, 109, 116, 117, 151, 152, 179, 199, 203, 230, 234, 238 univariate beta, 83, 106, 164 univariate Cauchy, 23, 56, 68 univariate chi-squared, 2, 21, 53, 63, 70, 80, 82, 83, 87, 90, 91, 98, 100, 103, 108, 120, 131, 132, 135, 152, 165, 181, 196, 200-202, 209, 216, 234, 239, 245

univariate Poisson, 93 univariate uniform, 226 Wishart, 193, 213 econometrics, 56, 114 Edgeworth form, 133 EM algorithm, 210-212, 235, 239 entropy, 21-23, 204 entropy loss function, 219, 220 error function, 141 exchangeability, 172 exponential family, 96

factor analysis, 193, 215 fiducial distribution, 45 Fisher's optimal discrimination criterion, 244 forecast error, 13 forecasting, 12, 15 Fortran, 158, 169 Fourier series expansion, 242 Fourier transform, 140

Gauss hypergeometric function, 46, 67, 78, 81, 137

Index Gauss-Hermite quadrature, 134, 143 Gauss-Legendre quadrature, 152 general hypergeometric function, 113 generalized gamma function, 113, 193 Gram-Charlier expansion, 133 Hall's index, 242 Hartley's differential-difference

equation, 133 Hermite polynomial, 132, 136, 176 hierarchical models, 214 Hotelling's T2 statistic, 199, 202, 203, 209, 210

271

missing data imputation, 212 moment generating function, 76, 121-123, 125, 198, 205

monotone data augmentation, 213 Monte Carlo algorithm, 152, 157 Monte Carlo simulation, 161, 162 multiple comparison procedure, 154, 158 multiple correlation coefficient, 91 multiple decision problem, 245 multiple regression analysis, 91 multiple stationary normal time series, 120

multivariate t model, 191-194, 219, 221, 222

importance sampling, 161, 163, 164 imputation multiple, 212, 213 single, 212

incomplete beta function, 62, 128, 139,

generalized, 196, 222 multivariate t statistic, 198, 199 multivariate skewness, 102 multivariate tail-dependence, 243 mutual information, 24, 25

148, 149

incomplete beta function ratio, 139, 232 incomplete gamma function ratio, 47 infinitely divisible, 41, 43 information matrix, 217-219, 235 inversion theorem, 135

nonlinear model, 239, 240 nonlinear regression, 235, 239 multivariate, 235 normal law, 198, 199

Kullback-Leibler number, 24, 204, 205 kurtosis, 28, 121-123, 125, 126, 200, 223, 224

online environmental learning, 246 order statistics, 186, 207, 208 orthant symmetry, 202 orthogonal expansion index, 242 orthogonally invariant, 97, 117

Levy representation, 41, 43 Laguerre polynomial, 177 Laplacian T-approximation, 214 Laplacian approximation, 214 lattice rule algorithm, 157, 158 least absolute deviation estimator, 243 linear inequalities, 158 linear model Bayesian, 102, 233 classical, 228, 239 general, 237 indexed, 235 linear regression, 213 multivariate, 213 linear simultaneous equation model, 114 load, 158

parabolic cylinder function, 90, 144 partial regression coefficient, 216 percentage points, 57, 59, 174-178, 180-185, 187-189 Poisson mixture, 93 portfolio optimization, 243 predictive distribution, 114, 232, 233 prior diffuse, 50, 52, 114, 234 improper, 240 natural conjugate, 52, 114, 234 proper, 240 probability inequalities, 165, 169-172 projection index, 241, 242 projection pursuit, 241

logit, 164

MacDonald function, 36, 38, 40, 42 macroeconomic modeling, 114 Mahalanobis distance, 210

quadrant dependence, 20 quadratic form, 19, 20, 34, 50, 96, 100,

MANOVA, 239

104, 115, 125, 197, 198, 218 quadratic loss function, 194, 222 quadrature formulas, 137

maximum entropy characterization, 23 distribution, 23 maximum probability content, 160 microeconomic modeling, 114

Renyi distances, 28 Renyi information, 26-28 randomized block design, 178 ranking and selection, 56

Index

272 rejection algorithm, 226 reliability, 152 robust estimation, 235 Schur-concave, 2, 172 score function, 215, 217 score test, 215, 216 security returns, 246 Shannon entropy, 27 Simpson's rule, 169 skewness, 102, 121, 217, 223, 224 slope coefficient, 243 spectral decomposition, 195, 221

spectral density matrix, 120 speech recognition, 246 squared error loss function, 221 standardized cumulant, 133 StatLib, 212 Stein loss function, 219 stock market problems, 192 strength, 158 Student's t index, 242 Studentized maximum and minimum modulus tests, 139 Studentized statistics, 134 Taylor series expansion, 35, 230 multivariate, 154 triangular decomposition, 195 trigamma function, 235 Tukey's procedure, 154

variance component model, 214

Wishart matrix, 191-193, 196, 204, 219, 221, 222

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