COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI, 21.
ANAL)-TIC FUNCTION METHODS IN PROBABILITY THEORY Edited by:
B. GYIRES
NORTH·HOLLAND PUBLISHING COMPANY AMSTERDAM - OXFORD-NEW YORK
PREFACE
This book comprises the proceedings containing detailed versions of most of the papers presented at the Colloquium on the Methods of Complex Analysis in the Theory of Probability and Statistics held in the Kossuth L. University of Debrecen, Hungary from Augustus 29 to September 2,
1977 as well as some others which were
submitted later. All papers in this book were refereed. The Organizing Committee consisted of B. Gyires (chairman), P.
Bartfai
(secretary), L. Tar (secretary),
M. Aratb, P. Medgyessy, P.
R~v~sz,
K. Tandori, J. Tomkb,
I. Vincze. There were 49 participants at
t~e
Colloquium from
10 different countries, including 19 from abroad. I wish to thank Professor E. Lukacs for his suggestions throughout the organization of the Colloquium.
Thank is also due to Dr.
P.
Bartfai for taking
charge of the correspondence.
B. Gyires
-
3 -
ClJNTENTS
PREFACE
3
COHTENTS
5
SCIENTIFIC PROGRAII ..
7 I I
LIST OF PARTICIPANTS
P.
Bartfai, Characterizations by sufficient statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H.
Bergstrom, Representation of infinitely divisible probability
~easures in
~2
and some of its
subspaces. . .. . . . . . . . .. . . .. . .. . . . .. . . .. . . . . .. .. E.
Csore8 -
L.
Stacho, A step toward an
43
~symptotic
expansion fo( the Cramer - von l!ises statistic
z.
Daroczy -
W.
Eberl, Recursively defined l1arkov processes
53
Gy. iMaksa, Nonneeative information
functions ..
(discrete T.
21
Csaki, On so~e distribution concerning maximum and minimum of a Wiener prOcEss..............
S.
15
67
para~eter).......................
79
Gerstenkorn, Distribution of the sum and the mean of mixed random variables in a class of distributions . . . . . . . . . . .
z.
Govindarajulu -
A.P.
93
Gore, Locally most powerful
and other tests for independence in multivariate populations........................ B.
Gyir~s,
99
On a generalization of Stirline's numbers
of the first kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1L:3
B. Gyires, Constant reeression of quadratic statistics on the sample mean, W.K.
Hayman -
II . . . . . . . . . . . . . . . . . . .
I. Vincze, Uarkov-type inequalities
and entire functions . . . . . . . . . . . . . . . . . . . . . S.K. Katti,
137 153
Infinite divisibility of discrete
distributions,
III ..
165
5 -
S.K. Katti - J. Stith, An empirical graph for choosing a population distribution using a characterization property •..••.......••••.....
173
K. Lajko, A characterization of generalized normal and gamma distributions ....••.•.•.•...•.••....
199
E. LukAcs, On some properties of symmetric stable distributions . . . • . . . . . . . . • . . . . . • . . . . • . . . • . • . . . 227 J. Panaretos, A characterization of a general class of multivariate discrete distributions •..•.... 243 J. Panaretos - E. Xekalaki, A characteristic property of certain discrete distributions .... 253 Gy. Pap, On the asymptotic behaviour of the generalized binomial distributions •.••.•.....•...... 269 B. Ramachandran, On the strong Harkov property of the exponential laws ....•.....••••.•. , . . . . . . . . 277 B. Ramachandran, On some fundamental lemmas of Linnik •••••.....••.•.....••...••.•.•.......... 293 V.K. Rohatgi, Asymptotic expansions in a local limit theorem ••••........•.•..•..•....•..•••......•. 307 K. Sarkadi, Characterization and testing for normali ty •••••••..•..•...•..........•......... 317 V. Seshadri - G.P.H. Styan, Canonical correlations, rank additivity and characterization of multivariate norm§llity •••.••..•......•......•...... 331 F.W. Steutel, Infinite divisibility of mixtures of 345
gamma distribution S.J. Wolfe, Mixtures of infinitely divisible
distribution functions ..•....•...•..•....•.•.• 359 E. Xekalaki, On characterizing the bivariate Poisson, binomial and negative binomial ~
distributions •••••...••......••••.•..•....•.•. 369
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6 -
SCIENTIFIC PROGRAM
29. August 2.00 -
3.00 p.m. Opening
3.00 -
3.30 p.m. E. LukAcs: On some properties of symmetric stable distributions
3.30 -
4.00 p.m. H.
Bergstrom: Representations of
infinitely divisible measures in
Rk
and in the Hilbert space by Gaussian invariants 4.00 -
4.30 p.m. K.
Sarkadi: Characterization and
testing for normality 5.00 -
5.30 p m.
I. Vincze: On a probabilistic problem concerning
6.00 p.m.
5.30 -
e~ire
functions
S.J. Wolfe: ri1ixtures of infinitely divisible d~stribution functions
\
30. August Chairman: H. 9.00 -
9.30 a.m.
Bergstrom B. Gyires: Constant regression of quadratic statistics on linear statistics
9.30 -
10.00 a.m.
W. Eberl: Convergence of recursive defined Markov processes
10.00 -
10.30 a.m.
o.
GulyAs - G.
L~grAdy:
Sampling
theorems for homogeneous and isotropic random fields 11.00 -
I I .30 a.m.
B. Ramachandran: On the strong Markov property of the exponential laws
11.30 -
12.00 a.m. J. Panaretos! A characterization of a general class of multivariate discrete distributions
-
7 -
Chairman: 3.00 -
B.
Ramachandran
3.30 p.m.
S.K.
Katti: An empirical graph for
choosing a population using a characterization property 3.30 -
P.
4.00 p.m.
Bartfai: Characterizations by
sufficient statistics 4.00 -
F.W.
4.30 p.m.
Steutel: Mixtures of gamma
distributions Chairman: 5.00 -
I. Vincze D. Dugue:
5.30 p.m.
Characteristic functions
in analysis of variance and design of experiments 5.30 -
K.
6.00 p.m.
Lajk6: Char8cterizations of
generalized normal and gamma distributions 31. August Chairman: T. 9.00 -
9.30 a.m.
9.30 -
10.00 a.m.
Gerstenkorn V.K.
Rohatgi: Asymptotic expansions
in the central limit theorem S. Csorg8: On an asymptotic expansion for the Cramer von Mises statistic 10.00 -
10.30 a.m.
A.
Szep: Random power series with
weakly dependent coefficients Chairman: V.K. 11.00 -
11.30 a.m.
Rohatgi Gy.
Pap: On the asymptotic behaviour
of the generalized binomial-distributions
- 8 -
31.
August
11.30 -
12.00 a.m.
V.
Seshadri:
A theorem on cha-
racterizing the Chairman:
3.00 -
S.S.
law
Wolfe Z.
3.30 p.m.
~ormal
Daroczy:
Uber die Characterizierung
der Entropy
3.30 -
B.
4.00 p.m.
Forte:
Non-symmetric entropies and
random variables
4.00 -
E.
4.30 p.m.
Csaki:
On some distributions
concErning maximum and minimum of Wiener process Chairman:
5.00 -
V.
Seshadri
5.30 p.m.
V.M.
Zolotarev:
On representations
of mathematical expectations
5.30 -
6.00 p.m.
J.G.
Szekely:
(
On a Chernoff t)(pe
function
1.
September Excursion
2.
September Chairman:
V.M.
9.00 -
9.30 a.m.
9.30 -
10.00 a.m.
Zolotarev H.
Kac:
Some probabilistic aspects
of potential theory S.K.
Katti:
Infinite divisibility of
discrete distributions,
10.00 -
10.30 a.m.
E.
Xekalaki:
Part
III.
On characterizing the
bivariate Poisson binomial and negative binomial distributions
-
9 -
Chairman: E. 11.00 -
11.30 a.m.
Lukacs
B. Ramachandran: On some fundamental lemmas of Linnik
11.30 -
12.00 a.m.
T.
Gerstenhorn:
Distribution of the
sum and the mean of mixed random variables in a class of distributions 12.00 -
12.30 a.m.
M. Dewess: The tail of distribution functions and its connection with the growth of its characteristic function
10 -
\
LIST OF PARTICIPANTS ARAT6, M., Res.
lnst. for Applied Computer Sci.,
Csalog~ny u.
30-32, PL 227,
BARTFAI, P., Math. u.
13-15,
Inst.
1536 Budapest, Hungary
of Hung. Acad. Sci., Re<anoda
1053 Budapest, Hungary
BERGSTRHM, H., Dept. Math. Chalmers Univ. of Techn. and Univ. of Goteborg, 40220 Goteborg, Sweden BRUINS, M.E., Joh. Verhulststraat 185, Amsterdam-ZI, The Netherlands CSAKI, E., Hath. u.
Inst. of Hung. Acad.
Sci., ReAltanoda
13-15,1053 Budapest, Hungary
CSIK6s, M., Nyisztor t~r 4/b, 2100 Godollo, Hungary CSIszAu, I., Math. u.
lnst. of Hung. Acad. Sci., ReAltanoda
13-15,1053 Budapest, Hungary
CSHRGO, S., Bolyai lnst. Jbzsef A. University, Aradi v~rtanuk
tere I, 6722 Szeged, Hungary
DAR6CZY, Z., Hath.
lnst. Kossuth L. University, PL
12,
4010 Debrecen, Hungary DEWESS, UONIKA, Dept. Hath., Karl Uarx Univ.,
701 Leipzig
GDR DUGUE, D., 24 Rue Jean Louis Sinet, 92330 Sceaux, France EBERL,
~.,T.,
Fleyer Str.
ERTSEY, 1., Hath.
122 c, 5800 Hagen, GFR
lnst. Kossuth L. University, PL
12,
4010 Debrecen, Hungary FEUER, tVA, Res. Inst. for Appl. Computer Sci., CsalogAny u.
30-32, PL 227,1536 Budapest, Hungary
FORTE, B., Dept. of Appl. Hath., Univ. of Waterloo, Waterloo, Ontario, Canada GERSTENKORN, T., Math.
Inst., Univ. of Lodz, ul.
Inzyniezska 8, 93569 Lodz, Polen GLEVITZKY, G., Math.
Inst. Kossuth L. University, Pf.
4010 Debrecen, Hungary
-
1I -
12,
GULyAs, 0., Inst. of Meteorology, Kitaibel P. u.l,
1024
Budapest, Hungary GYIRES, B., lIath.
Inst. Kossuth L. University, Pi.
12,
4010 Debrecen, Hungary JELITAI, A., Munklsotthon u.
34,
1043 Budapest, Hungary
KAC, M., Rockefeller Univ., New York, NY 10021, USA KATTI, S.K., Univ. of Missouri, 314 Math.
Sci. Building,
Columbia, HI 65201, USA KRAMLI, A., Res.
Inst.
Victor Hugo u. LAJK6, K., Hath.
for Compo and Automat.
24,
Sci.,
1132 Budapest, Hungary
Inst. Kossuth L. University, Pi.
12,
4010 Debrecen, Hungary LUKACS, E., 3727 Van Ness Str. NW, Washington, DC 2016, USA NEUETZ, T., Hath. u.
13-15,
Inst. of Hung. Acad.
Sci., Reliltanoda
1053 Budapest, Hungary
PANARETOS, J., 8 Gratesideias Str., Athens 504, Greece PAP, GY., Math.
Inst. Kossuth L. University, Pi.
12,
4010 Debrecen, Hungary PINCUS, R., Zentralinst. 39,
fur Hath. und Hech., Hohrenstr.
108 Berlin, GDR
PROHLE, T., Res.
Inst. for Appl. Computer Sci.,
Csalogliny u.
30-32, Pf.
RAISZ, P., VHrHsmarty u.
227,
1536 Budapest, Hungary
27, 3530 Hiskolc, Hungary
RAMACHANDRAN, B., Indian Stat.
Inst.,
7 SJSS Uarg, New
Delhi 110029, India REINITZ, JULIANNA, Hunklicsy u. RtVtSZ, P., Hath. u.
13-15,
I, 5350 l1iskolc, Hungary
Inst. of Hung. Acad.
Sci., Reliltanoda
1053 Budapest, Hungary
ROHATGI, V.K., Dept. Math., Bowling Green State Univ., Bowling Green, OH 43403, USA SARKADI, K., Hath. u.
13-15,
Inst. of Hung. Acad.
1053 Budapest, Hungary
-
12 -
Sci., Reliltanoda
SESHADRI, V., Dept. Math., McGill Univ., S05 Sherbrooke St. West, Montreal, H3A2K6 Canada SPIEGEL, G., National Planning Office, Angol u. 27, 1149 Budapest, Hungary STEUTEL, F.W., Dept. Math., Eindhoven Univ. of Technology P.O.Box 513, Eindhoven, The Netherlands SZtKELY, J.G., Hath. Inst. Eotvos L. University, Ufizeum krt. 6-S,
lOSS Budapest, Hungary
SZENTE, J., Res. lnst. for Appl. Computer Sci., CsalogAny u. 30-32, Pf. 227,1536 Budapest, Hungary
sztp,
A., Math. Inst. of Hung. Acad. Sci., ReAltanoda u.
13-15, 1053 Budapest, Hungary
TAR, L., Math. Inst. Kossuth L. Univ., Pf.
12,4010
Debrecen, Hungary TOMKO, J., Uath. lnst. Kossuth L. Univ., Pi.
12,4010
Debrecen, Hungary VlNCZE, I., Hath. lnst. of Hung. Acad. Sci., ReAltanoda u.
13-15, 1053 Budapest, Hungary
VIGASSY, J., Central Inst. for Phys. Res., Konkoly Thege fit, Pf. 49, 1525 Budapest, Hungary WOLFE, S.J., Dept. Hath. Univ. of Delaware, 501 Kirkbridge Office Building, Newark, Delaware 19711, USA XEKALAKI, EVDOKIA, Paxon Str.
IS, Athens 812, Greece
ZOLOTAREV, V.U., Steklov lnst. of Acad. Sci. of USSR, Vavi lova 42,
117333 Moscow, USSR
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COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
CHARACTERIZATIONS BY SUFFICIENT STATISTICS P.
BARTFAI
INTRODUCTION The result stating that the sample mean is a sufficient statistic for the location parameter of a family F(x-0)
of distribution functions only if
normal,
is well-known, but till now it has been proved
F(x)
is
only under strong restrictions. For e.g. KAGAN-LINNIK-RAO ([IJ, Theorems 8.5.3 and 8.5.4) assume the existence of the density of the sample mean. Now we shall show that this statement as well as the analogous statement for the scale parameter is true without any condition. I.
THE MEAN IS SUFFICIENT STATISTIC FOR THE
LOCATION PARAMETER THEOREM I. Let distribution fU1lction
X 1 'X 2 " " ' X n F(x-O).
e
sufficient statistic for
-
Then
iff
15 -
be i.i.d.
r.v.-s with
I ;:;(X I +X 2 + •• • +X n )
F(x)
is normal.
is
PROOF. Consider the n-dimensional sample space (Rn,B(n),P a ) where B(n) is the set of the n-dimensional Borel sets, P a distribution function Ea
is
the measure generated by the
F(X I -a)F(x 2 -a) ... F(Xn-a). Write
for the expectation with respect to x1+···+x n
Pa. Let
r(~)
and
n
c then the equality
Po-a.e. and h does not depend on it(T-a) e and taking an expectaa. MUltiplying it by
holds for every
a
tion we obtain ( 1)
The expectation on the left hand side does Rot depend on a
because the set
shift by a vector
C
is invariant with respect to a
(a,a, ... ,a)
and so
f (t) •
Therefore the right hand side of (I)
is equal to
f(t),
too, or, which is the same (2)
f(t). Introduce the distribution function
our aim (4) we can assume that
-
16 -
(considering
EO(lc) > 0)
(3)
it is really a distribution function because
f(O) f(O)
I.
The characteristic function of
f(t)
(2) ,
is, according to
therefore, using the Unicity Theorem of G 8 (z)
the characteristic functions,
i.e.
8,
=
G 8 (z)
cannot depend on
G(z).
Let us choose the value of W 1 +W 2 +· •• +wn
8
in (3)
8
=
it by
--
n
=
interchanged because
0 $ h $
1
a.e., and we get for
the inner integral that 00
••• J
f
E (E (I T
T
C
E (l c !8(W)+T)dF(W I ) ••• dF(W) T n !8(w)))
which implies
A
z
(~:
T(~)
< z},
we obtain
f ••. f (4)
=
(w l ,w 2 ' ••• ,w )ER) and integrate n dF(WI) ••• dF(W n ). The two integrals can be (w
n
Let
8 (w)
A
h(T)dF1 ••• dF n
z
-
17 -
Consider the r.v.-s
that to
X I 'X 2 ' ••• 'X n again (4) means XI+ ••• +Xn are independent with respect PO. This relation leads us to the well-known func-
X2 -X I
and
~(t)
tional equation for the characteristic function of
XI
~(t+s)~(t-s)
2
(t)
from which we can easily deduce that
~(t)
is the
characteristic function of the normal distribution. 2. THE MEAN IS SUFFICIENT STATISTIC FOR THE SCALE PARAMETER THEOREM 2. Let
X I 'X 2 ' ••• 'X n
distribution function F(+O) = O.
Then
statistic for
PROOF. set
F(~)
(9
be i.i.d. r.v.-s with
> 0).
Suppose that
I
-(X + ••• +X) (n ~ 2) is sufficient n I n 9 i f f XI has a r-distribution.
It is quite similar to the earlier. Let the
be defined by
C
c
XI>O, ••• ,Xn>O}
and introduce the distribution function (3' )
We can prove by the same argument that
'"' P"I (T < (4' )
z)
and
G(Z)P I (c)
PI ( {T <
-
z} c) •
18 -
G 9 (z)
=
G(z)
=
(4') means that
X 2 +···+Xn
independent with respect to
and using a theorem
PI
of LUKACS [2J we obtain the statement of Theorem 2. REMARK. Let
be a sufficient statistic for F(~)
the scale parameter of the family which is a natural assumption that a. eous function, i.e. T(A~) A T(~) of generality we can take
a.
=
8 T(X)
(8
X2
Xn
1
1
0). Assume,
is a homogen-
and without loss
1. By this method we can
prove that in this case
T(O, x' ... 'x)
>
and
are independent. A similar remark can
be made to the Section 1 too. REFERENCES [IJ" A.M. Kagan - Ju.V. Linnik -
S.R. Rao, Characteriza-
tion problems in mathematical statistics
Russian), Nauka, Hoszkva, [2J
(in
1972.
E. Lukacs, A characterization of the gamma distribution, Ann. Math.
Statist.,
26(1955), 319-324.
P. B~rtfai Math. lnst. of Hung. Acad. Re!ltanoda u.I3-I5,
Sci.
1053 Budapest, Hungary
-
19 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION lIETHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
REPRESENTATION OF INFINITELY DIVISIBLE PROBABILITY MEASURES IN t 2 AND SOME OF ITS SUBSPACES H. BERGSTROM
1. WEAK CONVERGENCE OF MEASURES The purpose of this paper is to describe infinitely divisible probability measures in t 2 by the projected measures on the line and to do this by a direct application of A.D. ALEXANDROFF's important framework [IJ. Consequently we rely upon linear functionals.
In contrast
to the main subject of the conference where this paper is presented, our analyses is pure real. ALEXANDROFF deals with measures and even signed measures on normal spaces. By a measure he means a finitely additive non-negative set function but we shall here only apply his theorems to
a-smooth (= a-additive)
measures and particularly to probability measures. Let V
be the class of bounded continuous functions from a
a-topological space
S
Sand
the algebra generated by
the closed sets. The first Alexandroff theorem, being a generalization of a famous theorem of F. RIESZ, states that a bounded non-negative linear functional V
~
determines a finite measure -
21
-
on
S
L(f)
on
(we simply say
on
such that
S)
J
L (f)
( 1. 1 )
(fEIJ') •
f(x))1(dx)
Weak convergence of a sequence
of measures on
S,
means convergence of
( 1. 2)
L
n
n
(dx)
IJ', determining a measure
on
holds.
f(x))1
and hence to a bounded linear functional
for any L (f)
J
(f)
We say that
such that
)l
converges weakly to
{)1n}
(I. I)
A sec-
)1.
ond fundamental theorem of Alexandroff tells us that the weak limit
is
)1
a-smooth if
are
)1n'S
the normal space is completely normal.
a-smooth and if
We deal here only
with metric spaces being such spaces completely normal. In a metric space the measure is determined by
IJ'o
considered on the class
(1.2).
of bounded uniformly contin-
uous functions and also for weak convergence it is sufficient to deal with
and
Ln(f)
fEIJ'O.
only for
L(f)
In many situation the weak convergence of a sequence of measures on a normal space
is determined by
S
corresponding weak convergence on a sequence of subspaces, which in a certain way approximate
S.
We shall here
consider such a situation in a special case.
THEOREM I. p
and
space
nCr) s(r)
a
I.
Let
S
be a metric space with metric
continuous mapping of
such that
continuous mapping
s(r)
vCr)
for
suppose that the mapping
x .... p(x,v(r)n(r)x)
-
S
onto a metric
is mapped into
22 -
S
by a
r= 1,2, . . . . Further
of
S
into
to
0
as
R r -
measures on on
p(x,v(r)n(r)x)
is continuous and Then a sequence
+00.
converges weakly to a
S
of
tends a-smooth
a-smooth measure
i f and only i f
S
(i)
the sequence
on
converges weakly to a measure
A(r)
for any
r= 1,2, ...
lim lim sup ~ {x:p(x,v(r)n(r)x) n n ..... oo
(ii)
>
£}
o
r .... oo
for any (iii)
sup n
~
n
£
(S)
> 0, < +00.
REMARK. The conditions for the weak convergence of
n
(r)
-
x
and
PROOF. Let
p ( x, L
n
v(r)
n (r) x )
'1'0.
are only measurable.
be the realvalued bounded linear
functional corresponding to class
(i)-(iii) are sufficient 1'kn also if the mappings ~n
~n
and considered on the
By
( 1 .3)
a bounded linear functional is defined. By the transformation (I. 4)
x=n
(r)
x
we get
L (r) (f)
n
Now assume that (i)-(iii) hold. Then by (i) converges to a bounded linear functional '1'0
as
n -
+00.
Using the form (1.4) for
further get
-
23 -
L(r)(f)
L(r)~f)
on
L~r)(f)
we
IL n (f)-L(r)(f)! n
I
(). 5)
S
sup
If(x)-f(y) !~n(S)+2 suplf(x)
p(X,y)SE
xEs
J
1
p
«r) r
x,v
1t
» x
The second term on the right hand side tends to n -
r
+00,
-
+00,
~n(dx). E
as
0
according to (ii). The first term is
arbitrarily small for sufficiently small
since
E,
is
f
uniformly continuous. Hence
o.
lim lim suplL (f)-L(r)(f)1 n n r ....co n .... oo
( I .6)
We now consider
IL
(r ) (r ) 1 (f)-L 2
(\.7) +IL
(r ) l(f)_L
Inn
Using the fact that n -
+00
L(r)
(f)
(£)1
S
IL
(r
(f)I+IL (f)- L
n
L (r) (f)
) 1
(r (f)-L
n
(f)
1
+
(r ) (r ) (r ) 2 (f)I+IL 2 (f)-L 2 (f) 1
n
n
L (r) (f)
converges to
n
) 1
as
and also using (1.6), we find by (1.7) that is Cauchy covergent and hence convergent lim
L(r)(f)
=
L(f)
r ... +oo
where
L
is a bounded linear functional on ~
hence determines a measure find that +00 {~} 1 n n=
{L(n) (f)}
on ~.
and
S. By (1.6) we than
converges to
converges weakly to
~O
L(f)
and thus that
This measure is a-smooth
according to Alexandroff's second theorem since the are
~n
a-smooth. Before proving the necessity of the conditions (i)-
-(iii), we verify the statement in the remark. Clearly, so far we have only required measurability of the -
24 -
mappings
1l
(r)
and
p(x,v(r)ll(r)x). Hence the state-
x -
ment in the remark is true. We now prove the necessity of the conditions
(i)-
-(iii) under the conditions on the mappings given there. Since
1l
(r)
+00 {].In} n= 1
Hence
is continuous the weak convergence of
. l '1es t h e wea k 1mp
(i)
is necessary.
convergence
is equal to
on
0
to the
Jg £ [p(x,v r
1l
(r)
Jg £ [p(x,v r Now
1l
0
[£+00],
x)11l
(r)
n
s
and the converimplies
(dx)
x)11l(dx).
(r -
as
£<0
is a
£
r -
ina ted convergence theorem. least equal to
on
a-smooth measure
p(x,V ll(r)(x) - 0
r tegral tends to
1
which
R
g [p(x,v(r)ll(r)x)]
Then
realvalued continuous function from gence of
into
R
equa 1 to
(-00,0)
[£, I]
). ].In « 1l r ) - I'
f
Clearly (iii) is necessary. Let
be the continuous function from
and linear on
0
+00) +00,
and thus the last inaccording to the dom-
But the first integral is at
].In({x: p (x,v(r)ll(r)x) > £}). Hence (ii)
is necessary. 2.
INFINITELY DIVISIBLE MEASURES ON
A
a-smooth measure
].I
R,
R(r)
AND
~2
on a metric space on which
convolution of measures is defined is called infinitely divisible if there to any positive integer a-smooth measure
].Ir
such that
"'r ].I=].Ir
r
where
exists a
,"r ].Ir
denotes the r-fold convolution of ].I. The infinitely r divisible measure is essentially characterized by this property. Let us consider the special cases and
n N
2.
+00 {].In n}n=1 ;ok
In
R, R(r)
[2] , p 101 h . , I h ave sown t h at a sequence of convolution powers of probability measures -
25 -
(hence
a-smooth measures) on
converges weakly for
R
an increasing sequence
{k }
ing to
+00
if and only if
(2. I)
{k ¢ ( ..:...) ,', [Il - e] }
as
n -
of positive integers tend-
n
+00,
nan
converges uniformly to a function
a
>
O.
Il
(The weak limit
Il
and the parameter (a
>
a
>
r(a,Il,')
for any
is infinitely
n}
n
r(a,Il,')
divisible). We look upon ing on
{Il
of
,',k
as a function dependO. The class of funct-
Il
is uniquely determined by
0)
Il
and uniquely determines
and was therefore called the
Il. The Gaussian invariants from
Gaussian invariant of
an additive Abelian semi-group with the unit element
0
and this semi-group is isomorphic to the multiplicative
M
semi-group measures tion).
Il
(under convolution as multiplicative composi-
Indeed we have r[ a,
(2.2)
if
of infinitely divisible probability
,',v , .j
v
and
measures.
lJ
r(a,Il,' )+r[ a,v,·]
are infinitely divisible probability
It has shown in
[2J that
r[a,Il,·j
has the
representation
(2.3)
+
J
[cjJ
(x- y ) -6 (~) + .!
-00
where
q
is a
a
a
a
a-finite
¢'
(~)-y-] q (ay) a I +y 2
a-smooth measure on
R
such
that
J IxI Ixl SI
2 q(dx)
<
+00,
J
I xl >a -
26 -
q(dx)
<
+00 (a
>
0)
and tending to
a
number and
0
as
are determined by
8
]J=]J*n
]J ,
+00.
Further
is a real
n
according to
f ~ ]In(dy),
=
lim n n-+ oo
(2.4)
a -
a non-negative number and these numbers
I+y
2 lim n f y ]J (dy). n oo £-0+ n-+ iYI<£
a 2 = lim
8
To given
and
a-finite, a-smooth measure
properties stated above, iant
(2.3)
there exists a Gaussian invar-
belonging to an infinitely divisible probabil-
R .
ity measures on the vector
The projection
with norm
p
v
a-smooth measures on
are
11
p is defined by
and it is a continuous mapping of and
with the
q
belonging to 11
R(k)
onto
R(k)
and
p
= p·x
X
R. I f )..
=
]J
]J 1, V ,
it follows by the definition of convolutions that the -I ]J(ll-I.) and v(ll-I.) )..(11 .), on
projected measures R
p
p
p
satisfy the relation
).. (11
-I P
• )
]J(ll
-I
P
·),"V(ll
-I
P
.)
R =
where convolution is considered on
11
p
R(k).
Applying
Cramer-Wold's theorem we find that the infinitely divisible measures on
R(k)
are determined by the correspon-
ding projected probability measures.
To any such in-
finitely divisible measure there belongs a Gaussian invariant. We call the collection of these Gaussian invariants the Gaussian invariant of the infinitely divisible measure. We state THEOREM 2. I.
The Gaussian invariant of an infinitely
divisible probability measure collection of functions
]J
r(a,p,]J,·
-
27 -
on
R(k) (a
>
is the
0, p
vector
with norm
(i)
1)
r(a,P,Il,X)
where for
II
lim n J
(ii)
n- oo
(iii)
a
(i v)
q
2
P·Y2 lln(dy) , 1+11 yll
lim lim n J (p.y) 2 II (dy), lIyll:s£ n £-0+ n-+ oo
P
such that
J
a-finite measure on
a-smooth,
is a
II xII
2
>
O.
REMARK
1.
where
lIyll
q(dy)
<
J
+00,
in (ii) and
(iii)
2 p
+00
for any
Bp
The
is changed into
are not independent and the
Indeed we have
(i )
a
<
also has a representation,
r(a,P,Il,·)
are not independent.
~
q(dy)
II yll >£
REMARK 2.
m.
(k)
q({O})=O,
II y II :S 1 £
R
lim n J n ..... + oo k
~
k
~
y 2 lln(dy), 1+11 yll (i)
(j)
p p m .. , ~J
i=1 j=1 -
28 -
a
2 P
lim
moo ~J
To any given in (iv)
lim n
E .... O+
n ....+ oo
mo, mo ~
~
0
~J
0
and
with the properties
q
there exists an infinitely divisible probability ~
measure
with the invariants
r(a,p,~,·).
The proof of this theorem can be brought back to the proof of the corresponding theorem in
R. All we have to
prove is that the limits of the projected measures determine the measure
q
on
R(k). We have given a solution
on this problem in [2J p.320. However the proof given 2 there can be simplified. We now turn to the ~ -space. It is a Hilbert space and in fact any separable Hilbert space is isomorphic to
~2. This means that the cha-
racterization of infinitely divisible probability
~2
measures on
gives the corresponding characterization
~2
on any separable Hilbert space. The space space of all sequences
x =
+00
l. (x(j))2 <
with
+00.
{x(j)}~oo
J=I'
x(j)
is the real number,
We call these sequences vectors
j=1 and
x(j)the
j-th coordinate of
scalar product
goes from of
~2
(r)
=
II xii =x· x
~(r)
:x
~2
onto
The space has the
the norm j=1 p(x,y) = Ox-yU. The mapping
x·y
and the metric
1t
x.
x
1t
(r)
X
onto R(r) and is called the projection R(r) . It is easy to show that it is
continuous. Let
x(r)
the j-th coordinate other coordinates
be the element in
( 0) x J
o.
for
j=I,2, ... ,r
Then clearly
-
29 -
~2
which has and all
R(r}
is a continuous mapping of x -
V
and
(r)
~
x -
(r).
(r)
~
(r)
+~. Putting
.
xU
=
6 x r
and hence 2 2 1 into 1
.
a cont1nuous mapp1ng of
Ux-v(r}~(r}xU
R. Further
into
.
1S a cont1nuous mapp1ng of
X
UX-V
12
into
.
tends to
0
as
2
r
Ux_v(r}~(r}xn
we have
x_i(r}
1
116 xU. r
The space
12
has all properties required in
Theorem 1.1, which then may be applied when we deal with weak convergence in If
is an infinitely divisible probability
II
12
measure on
lln
12.
then
with a probability measure
for any positive integer
n. We shall essentially
use this property in order to characterize
{ll:k n }, where
all at once we consider sequences
is a sequence of probability measures and uence of positive integers tending to
kn
k
n
+~
k
n as
{lln}
a seqn
t
+~.
be a probability measure on
THEOREM 2.2. Let
and
ll. But then
,2,... and let { ll "'k n } n. Then n
a positive integer for
tend increasing to
with
+~
n=1
converges weakly to a probability measure
on
II
i f and only i f ·'·k
(i)
{ll (~(r}-l.}}" n converges (r) n of 12 weakly for any projection ~
the sequence
into (i i)
lim
R(r} lim
(r=I,2, ••. ),
k
r .... +co n .... + co
(ii i) lim
(i v)
n
f
II
n6 xii >e: r
n
(dx)
0,
r .... + oo
lim sup k 6 x II (dx) n f n r n n .... + oo 116 xiISe: r
lim
lim sup k
r .... + oo
n ....+ oo
n f
116 xl r
2
II
n
(dx)
0,
O.
For the proof of this theorem we need several lemmas.
-
30 -
LEMMA 2.1. Let
~2
OIl
such that
A=~*V. If
A({x:llt. xII
>
r
then there exists ~{x:llt.
r
We may choose etrical,
x
r
>
xU
and assume that 1-e: }. Then r
°<
E
r
r
r
r
t. (x+x
) r
is symm-
~
t. x =x • If r
e:
and
>
1-e:
put
r
.
r
r
f~(K-x)v(dx)
A(K)
>
0,
xr· If we get
if and only if
{x:Ut. xII ~ e:} r E = {x:~(K-x»
V(E c
<
)
~
>
1, i.e. vee)
0, which
is not empty. Thus we may choose
> 1-e: r , i.e. ) = t. x+t. x • r r r
=
K =
Further put
and then by the definition of
r
-K=K
<
1-e: r
tells us that
t.rxr
e: .
is a measurable set and we get
E
e:
r
1,
such that
<
e:}
be probability measures
< 2e: r .
e:)
A (K)
and hence for
~(K-x
<
r
such that
r
PROOF. To given
x EE
>
r
e:
~2
in
r
(x+x )R
x
<
e:})
v
then even ~(x:llt.
>
and
A,~
~({x:Ut.
r
(x+x )II r
we obtain
E
>
e:})
Hence we may choose
< xr
e: r
clearly
such that
is symmetrical, then, observing that = ~(K+x ). Further yEK-x nK+x
~
~(K-x)
r
lit.
r
r
~
(y+x )0 r
r
e:, Ut. (y-x )11 r
inequalities together with
-
31
-
r
~
r
e:. These
s imply
n6 r yn S {K-x
Since
].l(K-x
r
r
n6 r-y+x r n+n6 r y-x r n which means that
E,
}n{K+x }C{yq6
r
>
)
1-E
and
r
>
S E}
].l{y:n6 yn
r
r
yn S E}. ].l(K+x
1-2E
r
r
R
we hence have
r
.
We shall use the function x into
> 1-E
)
expC-nxn 2 ) from
12
and now give some properties of this function.
LEMMA 2.2.
We have
!exp(-nxn 2 )-expC-nx+tn 2 )-2x.t exp(-nxI1 2 )! S a(n)ntn 2
(i)
with· finite a(n) g(x,t)
=
ntn S n < +00. Further
for
2 2 2 2exp(-nxn )-exp(-nx-tn )-exp(-nx+tn )
satisfies the inequalities (ii)
!g(x,t)!
(iii)
g(x,t)
1
ntn S I
for
for
for
c 1 (n)
!g(x,t)1 S c 2 lit
(i v)
(v)
for all
x
and 1
t,
I xn S 7; nn tn
with
n>o,ntn 2n,
for
of
~
S 2
x
and
2
with a positive number
(independent
t), ~
g(x,t)
nxn S
I
±,
c3ntll
ntn S
(independent of
x
2
± and
with a positive number t).
- 32 -
c3
PROOF.
II xII
Observing that 2
-lIx+tll
IIItll2
for
II til
-to
=
::; n
we
get
2
-II til
-2x·t,
::; n 2 + 211xlln
+ 2x·tl
(i)
by the help of
exp-[ 1It11 2 +2x.t].
expansion of obvious and
2
(iv)
The
is obtained by
For the proof of
(iii)
the Taylor
inequality
(i)
regarded
(ii) for
is t
and
we write
2 2 exp (-lIxli )[ 2-exp(-1It1i +2Ix.t!)-
g(x,t) (2.5)
-exp (-II tIl 2 -2! x· t I)]
and observe that for
II xii
~ exp (
1
4"
::; 1
n,
II til II til
2
~
2,
-2!x.t!
i
Then by
n. 1
~
2
-T6'l )(2-2exp(- "2 n )) =
IItIl2+!x.t~ ~
n2
(2.5) we get
c 1 (n).
g(x,t)
For the proof of
we write
g(x,t)
By Taylor's
=
2exp{-CIixli
formula we
cosh(2!Xotl)-
and
get,
2
+11 til
for
2
)}[exp(lltli
II xii
::; -21 IItll2 cosh
thus
and
-
33 -
n2
::;
±, 1
"2 '
2
)-
II til
::; 2 '
~
(v)
LEMMA 2.3.
{A
Let
} +00
(r= 1 ,2, •.. )
n,r n=1 of probability measures on i 2 tending to
positive integers
lim sup k
lim
(i)
r--+ oo
n"'+ oo
lim sup k
lim
(ii)
r"'+ oo
n"'+ oo
lim sup k
lim
(iii)
r"'+ oo for
n"'+ oo
a
n -
+00.
as
+00
J
A
J
t.
n lit. yl>e: n,r r n lit. yiISe: r r
YA
sequence of If
0,
(dy)
(dy)O
0,
n,r
U,\ yl12A
J n lit. ylSe: r
(dy)
r
n,r
n(dy)
°
°
A
lit. yll>n n,r
n ..... + oo
r"'+ oo
"'k
J
lim sup
lim
r
n > 0.
any
PROOF. Let put
{k} n
e: > 0, then
any
(i v)
for
and
be sequences
*0 A n,r
e. We have the identity
e-A
(2.6)
be the unit probability measure and
e
"'k
k
n
n,r
n !:
-I
j=O
A'~j (e-A n,r
n,r
).
By this identity and the properties of convolutions we get, applying (i) of Lemma 2.2 i'k
J[ l-exp(-IIt. r Y112)] An, n(dy) S r -I
k n
!:
j=O
suplf{exp(-IIt. xii
2
r
I
-
34 -
2 )-exp(-IIt. (x+y) II )}A r
n,r
(dy)l:::;
S k
f
nll6. yll>n
A
r
n,r
+ 2k II f 6. y A Cdy) II n 116. yllSn r n-r
(dy)
+
r
+ k a(n)fn6.
n
r
yR 2 An,r (dy).
By (i)-(iii) the right hand side of the last inequality
0
tends to (iv)
n -
as
+~
and
r
+~
-
in this order. Then
follows by the right hand side of the first inequal-
ity. Applying Lemma 2.3 and Theorem 1.1 {~n}
to the sequence
of probability measures we find that the conditions
(i)-(iv)
in Theorem 2.2 are sufficient for the weak ":k n convergence of ~n We have already remarked that (i) is necessary for this covergence. that also
It remains to prove
(ii)-(iv) are necessary conditions. We shall
first prove this for symmetrical probability measures ~
. Using the identity (2.6) we get according to the n symmetry of ~n
f exp(-II6. (x-t)1I2)[e(dt)-~ r
':k
n
n(dt)]
f{[exp(-II6. xII 2 )-exp(-II6. (x-t)1I 2 )lll "'k n(dt) r r n 2
f{exp(-IIL\ xii r
1 -2
f
g(6. x,6. r r
with the function
g
2
)-exp(-II6. (x+t) II r
,~k
n
n(dt)
t)~ n (dt) considered in Lemma 2.3.
Hence forming the convolution of the signed measure
)}~
*k n
e-~n
exp(-I6. x1l2) r
and applying (2.6) we get
the identity (2.7)
f
g(6. x,6. t)~ r
r
*k kn-I n(dt) = ~ n j-O
f
f
35 -
h. (6. x,6. t) 11 J
r
r
n
(dt)
with
with
h. (!:J. x,!:J.
(2.8)
]
We use
r
r
t)
= fg[!:J.
the identity
zeroelement.
(2.7)
r
!:J. t] / ' j (dy). r n
(x-y),
in the case when
x
By our assumption the sequence
converges weakly.
Then by Theorem
lim r-++ oo
0
to
r
as
Hp.nce
lim sup lln n ({x: II !:J.rxll n ..... + oo
r > O.
for any
for a
have the
-
We observe that
A
if
+00
A
sequence
lim sup
A({x:lI!:J. xII r
/'j({x:lI[\ xII
n
r=r(£)
clearly r=r(£)).
n
r
£ > 0
E)
= 0
for any
there exists
r
> n
and all
(2.10)
holds
<
E})
(r(£)
for
r
< 4'"
£
>
such
r=r(£)
(2.9) implies
2£
depends
> r(£)
on
£
only,
if it holds
for
Using these inequalities and applying Lemma 2.2
we obtain the following estimations. 1 1 £
tends
of probability measures we
n
A ({x:lI!:J. xII > c) < £ for all n. n r Then applying Lemma 2.1 we find that
(2. 10)
> £)
is any probability measure.
A ({x:lI!:J. xII>
if and only if to any
for
o
> E})
implications:
lim
that
n
1.1
"'k (2.9)
is the {ll "'k n}
(n $
lI!:J.rtll
2)
h.(O,!:J.t)~CI(") ]
For
r
f
/'j(dt)-2
II!:J. yli $£ n r
-
36 -
f
llj(dt)~
II!:J. yll >£ n r
~
".
0
if
E
is sufficiently small
large).
U~rtU
For
(2.7) 2
k
+ -
,',k
)]ll
r
n
n -
as
r -
+00,
+00
II n{x:U~ xU ~ E} then tends n r E > O. Thus we find that the conditions
for any (iv)
0
,"k
(repeated limit), since (ii) and
n(dt)~
n 2
The left hand side tends to 0
correspondingly
E. Using these estimations we
2J[ I-exp(-II~ til
to
r
(n S I)'
S n
for sufficiently small obtain by
(and I
in Theorem 2.2 are satisfied.
is obvious since
Clearly (iii)
is symmetrical.
We shall now remove the restriction that the are symmetrical. Then let in the sense
)j (E)=ll (-E)
in
II ;')j
J/,2. Then
n
for any measurable set
n
n
is symmetrical. Let
n
,',k
*
~
,',k
n = (ll
weakly to
ll.
weakly to
ll*)j. Hence the sequence
the conditions (2. I I)
lim r-'+ oo
(2. 12)
lim r-++ oo
Then
II
n
n
(ii) and lim sup k n-'+ oo
n
(iv)
n
*
,',k
{ll
n ,',k
ll) n
{ll *)j}
n n in Theorem 2.2,
J
II ",)j (dy) n n
J
II~
nn~ yll>E
n}
n
exists
converges
0
r
lim sup k n n-++ oo
r=r(E)
E
converge
satisfies
Ut:.ryIISE
r
xii
2
II ;'ll (dy) n n
The relation (2. II) holds if pnd only if to any there
lln
be the "conjugate" of
such that
\
-
37 -
O.
E > 0
(2. 13)
f ~ *~ 06 yO>£ n n
<
(dy)
for all
k£
n
r
n.
Applying Lemma 2.2 we conclude that vectors
v
chosen such that the measures
n,r
=~
n
(.
x
-x
n,r
may be n,r ) satisfy
the relation
f
v
06 xO>£ n,r
<
(dx)
r
and
(hence also for
r=r(£)
may be chosen such that (2. 14)
lim x ..... +QO
for any
m
n,r
6 x r
>
r n,r
=x
n
r(£». Note that
n,r
x
n,r
Hence
lim sup k f v (dx) n_+oo n06 xO>£ n,r
o
r
> o.
£
for all
£
kn
Put
f 6 xv (dx), 06 xO:s:£ r n,r r
A
n,r
For any
v
n,r
(. +m
£' (0 < £' < £) Om
n,r
O:s:
n,r
)•
we get
f
06 xn:s:£'
6 xv r
r
n,r
(dx)O+£
f
r
Hence (2.14) implies (2.15)
lim
lim sup Om
n,r
n
O.
Then it follows by (2.14) (2. 16)
lim
lim sup k
n
f
n6 xD>£ , r
Further -
38 -
v
(dx)
n,r
v
06 xO>£' n,r
O.
(dx).
II J t. r lit. xII::;£: r
X" n,r (dx)1I
t. xv (dx+m ) II II J r n,r n,r lit. xII::;£: r
II J (t. y-m )v (dy)lI::; lit. y-m II::;£: r n,r n,r r n,r
::;
J
lit. yll::;£: r
t. yv (dy)-m 11+ r n,r n,r
+2(£:+lIm
J
II)
n,r Note that the first
lim
lim sup k
r-+ oo
n-+ oo
Observing that
o
"
n,r
Hence
*"
n,r
=p *p n n
r-+oo
Regarding
J
J J nllt. (x)+t. yll::;£: r r
~ lim sup lim sup
(2.16)
n-+oo
and
(2.15)
we obtain by
n lit. xll<£: r
lim lim sup k
and
X" n,r (dx)1I
r
n-+oo
n-+oo
(2.14)
J t. nllt. xII::;£: r
lim lim sup k r-+oo
r-+ oo
(dy).
n,r
term on the right hand side of the
last inequality is zero.
(2.17)
v
lit. yll>£:-lIm II r n,r
imply
O.
(2.12)
lit. xII 2" ,',"I (dx) r n,r n,r
lit. x+t. y1l2" (dx)"I (dy) r r n,r n,r
2k { J lit. xII 2" (dx)n II t.rxll4 r n, r
(2.17)
tion, holding true for any
\ -
we conclude that this rela£:
>
39 -
0,
implies
~
o.
(2.18)
By Lemma 2.3, the relations (2.16)-(2.18) imply ,';k
lim lim sup A n ({x:Ht. xii> E)) n,x x r ..... + oo
(2.19)
>
o
Land L be the realvalued n n, x bounded linear functionals corresponding to lln and for any
A
E
O. Let
respectively and considered on
p~~jection
n(x)
of
£2
onto II
n
R(x)
'1'0. For the we have
( n (x)-I • )
since As in the proof of Theorem 1.1 the measure II (n(x)-I.) determines a realvalued bounded linear funcn tional L(x)(f) on '1'0. According to Theorem I. I and the proof of this theorem we have lim lim sup
x-+~
n-+~
(f) -L (r) (f) 1
1L
n
O.
n
Applying the inequality (1.5) to the difference
-L(x)(f) n
and regarding (2.14) we obtain lim lim sup
x-+~
n-+~
!L
o.
(f) -L (x) (f) 1
n,x
n
The two last inequalities give lim lim sup r~+oo
n ..... +co
IL n (f)-L n, x (f)1
(2.20) lim lim sup r .... +oo
n ..... + oo
IL(f)-L n,x (f)1
- 40 -
o
L
n,x
(£)-
is the functional belonging to the
L (f)
where measure
being the weak limit of
II ,
"'k
= lln n{.+k
n
"'k n II
(m
n,r
-x
n,r
)}.
converges weakly to
n
But according to as
n -
+00,
-x
U
r -
a-smooth ,',k
A n = Now n,r A"·'·k n (2.20) n,r
+00.
This is only
possible if
(2.21)
lim lim sup k
n
Um
If this condition holds,
n,r
n,r
o.
then it follows
this relation holds true if we change i.e.
for
lln
we find that
from
>
E
An,r =ll n
that
that
into lln' n,r Proceeding then as above
instead of 'J n,r (2.21) holds with
Um
n,r
A
i. e.
U =0
lim lim sup k U J I:::, Xll (dx) U n . . . . + oo n UI:::, XU$E r n r for any
(2.16)
o
O. At last we find by (2.18) applied to
lim lim sup k J UI:::, xU 2ll (dx) r-+ oo n-+ oo nUl:::, XU$E r n
0
r
>
O. Thus we have proved that the conditions
for any
E
(i)-(iv)
in Theorem 2.2 are necessary,
Consider now an infinitely divisible probability "'n for all positive inmeasure II on £2. Then ll=lln tegers to a
n.
It follows by Theorem 2.2 that
a-smooth
J
a-finite measure
UxU
2
q(dx)
< +00,
UxU $1
J
n xn >E
q(dx)
<
+00
r
41
-
q
on
2 £
nll
converges n
such that
for any
£
>
~
O. Further
projected measures.
is determined by its
It can easily be proved that the
Gaussian invariants of the projected measures for finite pr0i!ctions have the Gaussian representations given in Theorem 2.1 if we change
=
weak limit of
n~n'
{~n}
q
being the sequence of
~2, and consider
probability measures on elements in
into the corresponding
q
x
and
y
as
~2. The fact that the weak limit of a
convolution product
~
*k n
n in
~
2
is infinitely divisible
can be deduced from the well-known theorems about weak convergence of such products in
R. We have then to apply
Cramer-Wold's theorem. REFERENCES [I]
A.D. Alexandroff, Additive set-functions in abstract spaces, a) Mat. b) Mat.
Sbornik, 9(1941), 563-628, c) Mat.
13(1943), [2]
H.
Sbornik,
169-238.
Bergstrom, Limit Theorems for Convolutions,
Almqvist & Wiksell, New YorklLondon,
H.
Sbornik, 8(1940),307-348,
Stockholm, John Wiley & Sons,
1963.
Bergstrom
Dept. Math., Chalmers Univ. of Techn. and University of Goteborg 40220 Goteborg, Sweden
-
42 -
COLLOQUIA HATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
ON SOME DISTRIBUTIONS CONCERNING MAXIMUM AND MINIMUM OF A WIENER PROCESS E.
csAKI
1.
INTRODUCTION
Let
w(t)
be a standard Wiener process, w(O)=O
and put ( 1 • 1)
M+(t)
w(u),
max O~u~t
( 1 .2)
M- (t) = - min
w(u),
O~u~t
(1. 3)
M(t)
max
I w(u) I
max (M+ (t), M- (t».
O~u~t
It is known
(see e.8. RtNYI [8J), that the use of
theta functions and certain identities between them lends itself particularly well to investigate the distribution of
M(t). For theta functions and identities a standard
reference is e.g. MAGNUS-OBERHETTINGER [7J. The definition of theta functions used in this paper is given below. \- 43 -
00
(1.4)
~
1+2
-&O(V,T)
2
(-I)n e iltTn cos 2nltv,
n=1 00
( 1 .5)
( - I)n e iltT(n+I/2)2 S1n . (2 n+ I) ltV,
~
2
-&I(V,T)
n=O 00
( 1 .6)
2
-&2(V,T)
e iltT(n+I/2)2 co s (2 n+ I)
~
It
v
n=O 00
(1.7)
1+2
-&3(V,T)
~
e
iltTn
2 cos 2nltv.
n=1 In this paper we determine the joint distribution of
w(t):
the following characteristics of
(I .8)
R(t)
(1.9)
Q(t) R(t)
+ M (t) R"(tT
(0
<
t
<
00),
(0
<
t
<
00).
is called the range of the process
Q(t)
(O,t); the ratio
the interval
the relative maximum of
w(t)
on
may be termed as
w(t).
Similar investigations can be carried out for the Brownian bridge process
B(t). We consider the following
characteristics: (I. 10)
( 1. 1 1 )
( 1 • 12)
( 1 • 13)
+ MB
max
B (t) ,
O~t~1
MB
-
RB
+ MB + MB
QB
min B (t) , O~t~1
..
+ M B RB
-
44
-
Finally we investigate the growth rate of the lower
R(t)
limits of
Q(t), giving certain analogues of
and
theorems of CHI'NG [2J
and HIRSCH [6J.
2. JOINT AND MARGINAL DISTRIBUTIONS OF
RAND
Q
,
To determine the joint distribution of
Q(t), we start from that of e.g.
in FELLER [4J.
M+(t)
ale
J
2
{)2(v,
0
en -
lit
where
a
>
-
ale
0, b
J
M-(t), given
this can be expressed as
2ilt)dv
2 e 2 2 v e -2-
e
0
>
and
In terms of theta functions and using
certain identities given in [7J
(2. 1 )
and
R(t)
2
2
ie ) 211 dv,
(ve -\1 0 2lti
0, e=a+b.
A straightforward but tedious calculation leads to the following THEOREM I. P(R(t)
<
2v(l-v)
ufi,
~
n= 1
Q(t) < v) (-I)n-I n
2
n
2
+ 2 ~ (-I)nn(2
(nu)-
-v
n=1
(0
\ -
45 -
<
u
< 00,
0
< v <
I).
v=1
By putting of
R(t)
in (2.2) we obtain the distribution
determined by FELLER [3J
<
P(R(t)
(see also TAKAcs [IOJ):
uit)
(2.3)
~ (-I)nn(2~(nu)-~(u(n+I»-~(u(n-I»). n=1
2
An equivalent
P (R ( t ) <
exp~ession
u it)
for
(2.3) is given by
=
(2n+I)2 1t 2
(2.4)
~
8
I
2u 2
e
(- + 2
n=1
u
By letting
u
the distribution of
I
(2n+l)
P (O(t) <
v)
1t
tend to infinity in (2.2), we get OCt): 00
(2.5)
2 2)'
2v(l-v)
~ (-I~
n= I
n-I /
(0
<
v
<
I).
n-v
This can also be expressed by means of the logarithmic
derivative of the gamma function
(digamma function).
Let
(2.6)
q,(x)
=
d
dx log rex),
then the following identity holds: 00
~
n=1 (2.7) ltV
cosec 2v
-
46 -
ltV
-I
Now we turn to the case of Brownian bridge.
M;
joint distribution of by SMIRNOV [9J functions
(see also TAKAcs [IOJ).
<
B
b)
-
(2.8)
ill
e
\
--;
2c 2
a
has been determined In terms of theta
this can be expressed by
M
where
MB
and
The
>
0,
b
>
0,
=
-2a
2
3
~)
ill
II
lll2))'
a , -\}3 ( ;-
-
2 . 2
(2ac -\)--
2c
c=a+b.
The joint distribution of
R
and
B
is given by
THEOREM 2.
2v
(2.9)
2
2
+ L
n=1
2
n(n-I) e- 2u (n-v) _ L n-v
(0
-2u n
n (n+l)
n-tv
n=1
<
u
<
By putting
8 2 '" 2 -2u n - vu '" n e
n=1 2
e
-2u (n+v)
o <
00
v
2
<
I).
has been determ-
The marginal distribution of ined by GNEDENKO [5J.
2 2
2 2
I
(I-v) L 2 2 + 2(I-v) L e n=ln-v n=1
v=1
in
(2.9) one
obtains 2 2
(2. 10)
1+2
L (1_4n 2 u 2 )e- 2u n n=1
or using theta function identities an equivalent expre-
+
ssion is given by (2. I I)
<
P(R B
Letting
u)
--3-
....~
u
n=1
It
---z:z
2
n e
Q : B
< v) = 2v 2 (I-v)
B
2 2
tend to infinity in (2.9), one gets the
u
distribution of P(Q
n
lt 2 v'2"i
~
n= I
2 2 n -v
(2. 12)
(l-v)(I-ltv cotg (2.12)
(0
ltv)
<
v
<
I).
determines also the limiting distribution of
the statistic sup (F (x) -F (x)) (2.13)
where
F(x)
n
x
Qn
SUp(F (x)-F(x)) n x
is a continuous distribution function and
is the corresponding empirical distribution func-
F (x) n
tion. More precisely, the limit relation (2. 14)
lim P(Q
n
<
v)
holds true. 3.
OF
INTEGRAL TESTS CONCERNING THE LOWER LIMIT
R(t)
AND
Q (t)
Concerning the lower limit of
M(t),
a theorem of
CHUNG [2] adapted to the Wiener process says that (3. I)
P (M(t) < a
(t)
It,
t
..
00
according as -
48 -
i.o.)
or
0
1t
(3.2)
'"
f
1 2 ta (t)
2
8a 2 (t)
e
dt
< "'.
or
On the other hand, a theorem of HIRSCH [6J
adapted
to the Wiener process says that
(3.3)
<
P(M+(t)
a(t)ft , t ... '"
or
Lo.)
0
according as
(3.4)
f a(t) dt
'"
t
< "'.
or
A further result in this direction is given in [IJ,
M+(t)
where the joint behaviour of
been investigated. Assuming that
~
a(t)
a(t)
ain regularity conditions on
M-(t)
and and
has
b(t)
and cert-
b(t)
we have
shown that (3.5)
=
1
or
0
according as 1t
(3.6)
where
J c(t)
a (t)
tc 3 (t)
=
e
2
2c 2 (t)
dt
or
< '"
a(t)+b(t).
Using the results,
the formulae
(2.11) and
(2.12)
and similar arguments such as in [IJ the following theorem can be proved: THEOREM 3. Let
and
u(t)ftt",
as
u(t) > 0, v(t) > 0, u(t)+O, v(t)+O t ... '" then -
49 -
P(R(t)
(3.7)
<
u(t)lt,
t
-
i.o.)
00
or
0
according as 11
J
(3.8)
I
tu 4 (t)
2
2u 2 (t)
e
<
or
dt
00
and
P(Q(t)
(3.9)
<
vet),
t
-
i.o.)
00
or
0
according as 00
(3. 10)
J vet)
dt
00
t
<
or
00
It is easily seen that Theorem 3 implies the following law of the iterated logarithm:
(3.11)
I.
We note that (3.11) is also a consequence of and
(3.5)
(3.6).
Since Chung's results
(3.12)
P(lim inf M(t)
(3. I) and
Vlogl~g
t
(3.2) imply that
)
I,
t- oo
we have (3.13)
lim inf R(t) t- oo
with probability
-Vlog~og
t
I.
-
50 -
2 1 i min f t- oo
M ( t )
Vlog \0 g
t
REFERENCES [IJ
E. Csaki, On the lower limits of maxima and minima of Wiener process and partial sums, lichkeitstheorie verw.
[2J
Z.
Geb., 43(1978),
Wahrschein-
205-221.
K.L. Chung, On the maximum partial sums of sequences of independent random variables, Trans. Amer. Math. Soc., 64(1949),205-233.
[3J
W. Feller, The asymptotic distribution of the range of sums of independent random variables, Ann. Math. Statist.,
[4J
22(1951), 427-432.
W. Feller, An introduction to Probability Theory and Its Application, Vol.
[5J
II, Wiley, New York, 1966.
B.V. Gnedenko, Kriterien fUr die Unveranderlichkeit der Wahrscheinlichkeitsverteilung von zwei unabhangigen Stichprobenreihen, Math.
Nachr.,
12 (1954),
29-66. [6J
W.M. Hirsch, A strong law for the maximum cumulative sum of independent random variables, Comm. Appl. Math.,
[7J
18(1965),
Pure
109-127.
W. Magnus - F. Oberhettinger, Formeln und Satze fur die speziellen Funktionen der Mathematischen Physik,
Springer, Berlin, [8J
A.
R~nyi,
1943.
On the distribution function
Hungarian), Publ. Math.
Inst.
Hung.
L(z)
(in
Acad. Sci.,
2(1958),43-50. [9J
N.V. Smirnov, An estimate of divergence between empirical curves of a distribution in two independent samples 14.
(in Russian), Bull. MGU, 2(1939), 3-
[10J L. TakAcs, Remarks on random walk problems, Publ. Math.
Inst.
Hung.
Acad.
Sci.,
E. CsAki
Mathematical Institute of the Hungarian Academy of Sciences 1053 Budapest, ReAltanoda u.13-15 Hungary
- 52 -
2(1958),175-182.
COLLOQUIA MATHEHATICA SOCIETATIS JlNOS BOLYAI 21.
ANALYTIC FUNCTION METHODS DEBRECEN
IN PROBABILITY THEORY
(HUNGARY),
1977.
A STEP TOWARD AN ASYMPTOTIC EXPANSION FOR THE CRAHtR-VON HISES STATISTIC S.
CSORGO -
L.
STACHO
I.
INTRODUCTION ~s
The present note
a continuation of [IJ.
notation used there will be kept here. review these notations.
U 1 , ••• 'Un
r.v.-s uniformly distributed on
Let us
denote [0,1 J,
The
first
independent
and
(t)
F
n
2
empirical distribution function of this sample. wn 1 2 nJ (F ( t ) - t ) dt is the Cramer-von Mises statistic, V
f
n n
o
n
(x)
=
(t)
V(x)
P (w 2 < x)
is its
n
denotes be
the
distrib~tion
in [IJ,
the first
complete asymptotic expansion for transform of
a
natural number
(Theorem 3)
1
and then tried
of [IJ reads as
and positive number
\ -
53 -
As a
the Laplace-Stieltjes
in powers of
s
n
author has given a
to invert this expansion, without reaching the The last result
while 2 w . Let
distribution function of the
square integral of the Brownian bridge process. starting point
and
function,
the characteristic function of
(limiting)
the
E,
final
follows:
goal. For
[fl v (x)-v(x)
( I. 1 )
~
n
k=1
1 k
-
(-) 4k (x)+O(n
i (s +
1 ) +E
)+
n
4k
where the coefficient functions
are completely
specified by terms of expectations of certain functionals of the Wiener process and by the derivatives of
V(x)
and B'';(S £) n '
( 1 .2)
o( J I T
(t)
f
nt
Idt),
n
where 1
-(s+2) (s+4) T
n
T
n
{t: n
(s,£)
£
1
2(s+ I) ~
Thus,
}
n
in order to prove the asymptotic expansion
in question, it remained to prove that
B*(S £) n '
=
_ O(n-(s+I)/2+E ). Unfortunately, we still cannot estim-
ate
B* on this desirable way. All we can do now is a n first step in this estimation procedure, the result of
which will be another form of the remainder term in (I. I). This new form (derived in Sec.
2) lends itself
B'';
for further analysis better than of (1.2), and n results from the fact that is (exactly) entier 2n times continuously differentiable. The latter fact, in turn, is a consequence of a recurrent formula for the n-dimensional volume of the intersection of an sional
sim~lex
and an
ric formula (proved in
n-dimen-
n-dimensional ball. This geomet~ec.3)
is of independent interest
and provides a good hope to compile tables of exact -
54 -
distribution and percentage points for
V (x) n
and for
the distribution functions of similar statistics. Some notes concerning this is contained in Sec.4. 2. THE OTHER FORM OF
B'" (s
n
'
e:)
(n) (n) UI ' ••• 'U n •
Denote the ordered sample by
A
simple integration gives the well-known alternative form of our statistic (2. I )
w
~
2 n
k=1
2k-1 2 +-n 12n
(U (n) k
--)
~ wn2 <- ~ , hence I 2n 3 Vn (~) - . I Let Sn denote the simplex 12n -- 0 , Vn (~) 3 in the unit cube of {(xl,···,xn):O ~ xI ~ ... ~ xn ~ I} the n-dimensional real coordinate-space Rn. Put I 3 2n-1 Let B (c ,p(x» denote the cn (-2' n -2 n , .. ·'-2-)· n n n p (x) • and radius n-dimensional ball with center c From here it follows that
n
From (2. I) follows that V
=
(x)
where (y)
+
J •.• J
n!
n
S nB
n
p(x} =
V(x-
=max(O,y}.
If
n
(c
n
,p ( x ) )
dx l ·· .dx n ,
I~n)+.
y, Here, for a real number vol [.] stands for the n-dimenn
sional Lebesgue measure, then this means that (2.2 )
V
n
(x)
= n!vol [s nB (c ,p(x») n n n n
c Einterior(s ), n n
faces of
S
n
faces, equally
and the distance of I
from two
c n
Z-, and from the other n I n+3 - - . Thus, for 12n < x < --2-
is equally I
n-I V (x)
n 12n then, = n!vol [B (c ,p(x»), and, if n+3 < x < n+6 n n n 12n2 - 12n2 ' to get V (x)/n!, we have to subtract the volume of two
I2n
n
\
-
55 -
ball-hats from the volume of the ball. the situation becomes
extremly V (x). n
we do not know exactly
For larger
complicated, If
> n - 3'
x
x
this is why
t h en
(c ,p(x», and vol [5 ] ;y At the present n- n n n n stage we are interested only in smoothness properties of
S CB
V (x).
n
V (x)
It is quite clear that
is piecewise
n
analytic on the whole line. We can have trouble only n+3 n+6 at x = ----2 ' ----2 ' and at all further x's, for which I2n I2n the ball knocks against the (n-2)-,(n-3)-, ... , I-dimensional boundary of the simplex.
However,
in all these
exceptional points the function behaves quite well, since, from (2.2) and the Corollary to Lemma 10 in Section 3 it follows
LEUMA 9.
Vn (x)
is everywhere
[I]
times contin-
uously differentiable. A look at the exact results in case of
n=I,2,3
shows that this result is sharp. Now put
6=6(s,~)=~(s+2)(s+4), and let ~
1 s+ 1
be the smallest integer for which
n
2a, and set
be larger than
integrating by parts f
n
V
6(--2(k)
n times,
( a- 1 )
-~)+I
dk
(x)=~
dx
n/3
. (1) e~txv (x)dx I/I2n n
J
(t)
n/3
. () e~txv a (x)dx, I/I2n n
J
(it)a-I whence
If n (t)
I
n/3
J
Iv(a)(X)\dX=
I/I2n
-
56 -
n
a=a(s,~) ~
a.
V (x). n
Let Then,
Thus
I
f T
(t)
f
n
()O
Idt ::; On
t
n ::;
2 a-I
t
a
n
I 6(a-l) ::;
I
-
°
2 a-I
dt
f6 n
'2(5+1 )+£
nn
that is I
B'"
B"'(s
n
n
'
- -(5+1)+£ Q O(n 2 ). n
£)
In this way we have the following variant of Theorem 3 in
[I
J:
THEOREM 3'.
number
For any natural number
£
V (x)-v(x) n
=
[s/2J ~
I k
(-)
f
Iv(a) (x) Idx},
1/12n
a
depends only on
Thus,
+
n/3 ){I+
2
~k(x)
n
k= I
- ~(s+I)+£ + O(n
where
and positive
5
n
and
5
£.
to prove the Conjecture in [IJ,
i.e. the
complete asymptotic expansion in question, it would be enough to show that, tural number
a,
for an arbitrary (but fixed) na-
the sequence
Q
=
n/3
f
I V (a) (x) I dx
n 1/1 2n n is bounded. The first reasonable such a is
boundedness of the corresponding one-term expansion, and, between
49
would prove a
in particular, the
convergence. Quite certainly, and a general
a
a=49. The I
n
rate of
there is no difference if we want to solve the
problem, and this estimation question still seems to be not quite easy.
- 57 -
3. THE RECURRENT FORMULA Let
dist k (·,·) denote the Euclidean distance in m Rk. A function S:R {subsets of Rn} is said to be upper 6
>
for any £ > 0 x,yER m we have
semicontinuous if
0
such that for
there exist a
s(y)C{vERn:dist (v,S(x» < £} whenever dist (x,y) < 6. n m Also, it is said to be concave if for each 0 $ a $ m and x,yER, as(x)+()-a)S(y)~S(ax+()-a)y). and denote the origin of R n Let 0 = (0, ••• ,0) Bn
we write simply centered in
for the closed unit ball in cERn and a positive number
o. For
=
B (c,p) n
~,f'(~)
tion
f
c
and radius
p. For a
denotes the first derivative of a func-
and for
t, uERn, (t,
product. The following result number of the T
p
C+pB n
is then the ball with center real
Rn
u)
(with
stands for their inner n+)
replaced by the
(n-I)-dimensional faces) holds true if
is any (not necessarily bounded) convex polyhedron,
but for our purpose a simplex suffices. LEMMA 10. If
T
is any simplex in
Rn, and
-vol [TnB (c,p)], then one can find simplices n n n-) and constants Bn ,a) , ... ,a n+ ) ••• ,Tn +) in R with the functions
A
n
(p)
n-)
A
n-
)
,
.(p)=vol
~
n-
)[ T.npB ~
A (p)=
n
T), •••
1
so that
we have
=
(3. I)
Here the value of
A
n-
)
,
.(p)
~
equals to the
(n-)-
dimensional volume of the intersection between the i-th -
58 -
face of P
and the
T
(n-l)-dimensiona1 ball of radius
centered at the projection
the original
of the center
c.
~
of
c
n-dimensional ball on the supporting
(n-l)-dimensiona1 hyperplane of the
i-th face.
Furth-
ermore, dist
[
ex.
~
n
if
(c,c.), ~
c.-c ~
is a non-negative multiple of
-dist
n
if
(c,c.), ~
c.-c ~
u.
is the normal vector of the
~
pointing outward from
~
is a non-negative mUltiple of
where
u.
ui '
i-th face of
T,
At last,
T.
if
Bn
=[
D,
n
where
T
if
n
cET,
is the n-dimensional spatial angle of the n cone formed by the rays issued from c, having an T
intersection with if
T
of positive length. In particular, T then B =vo1 B n = n n
is an inner point of
c
(Ill) n /
r (~
+ 1) •
PROOF. Without any loss of generality we suppose that the
center of the ball is the origin, i.e. n
c=o,
Bn(C,P)=PB .
Let
uERn
be a unit vector and
KeRn
be a compact ,
convex set. Assume that the two supporting hyperplanes of K
which are ortogona1 to
with
K
of less than
seen that the function is differentiable and,
n-)
u
have intersection figures dimension. Then it is easily
f(~)=vo1n[Kn{tERn:(t,U)
for all
J - 59 -
~,f'(~)
=
:;;
0]
= vol that
nif
(j f.k)
I [ Kn ( t : ( t ,
U )
= E;}].
i t
f
0
11 0 W s d ire c t 1 y
u I ' . . . ,umER are unit vectors with such that the intersection figures of
those of its
supporting hyperplanes
to some of the dimensions,
U. ' ~
s
then for
A (E; . ) ~
= (t : (
t,
lying orthogonally
are of less then
(i=I, ••• ,m)
F:R m -
the function
= vol n [ KnA ( E; I ) n.
F ( E; I ' . . . , E; m )
(3.2)
where
Fro m her e
n
~
~
vol n _ I [ KnA ( E; I ) n.
defined by
. . nA ( E; n )] ,
(i=1 , . . . ,m), we have
E;.} eRn
:0;
U .)
R
n-I
. . nA ( E; i-I ) n
(3.3)
n (t
: ( t,
Now we in
(3.3)
U .) ~
.
cla~m
= E; .} ~
nA (E;.~+ I) n ... nA (E; n ) ]
that
t
h
e
.
part~al
..
der~vat~ves
are continuous on the whole
the notations
D.(E;I, . . . ,E; ]
=(t:( t , u / =E;j}
and
(i=I, . . . ,m).
)=A(E;.), m ] K(E;I"" ,E;m)=K,
Indeed,
Rm.
E·(E;I, .. ·,E;
] the
of
~
m fun ct ion
~ith )= 0F
~ ~
can be considered by
(3.3)
Lebesgue measure of the
as
the
intersection of
concave and upper semicontinuous ••• ,D.
~-
1('),
D.
(n-I)-dimensional
functions
in [2J)
by
set-valued D I ( · ) , · ..
I, . . . ,D
m
(·),E.(·) ~
and
K(·).
function on
the well-known Brunn-Minkowski theorem
one can write
of
~
u
<.,
vol n- I [ K ( E; I ' . . . , E; m ) nE ~. ( E; I ' . . . , E; m ) n
•
~
m n j= I
D.(E;I, . . . ,E; ]
OF
Thus if[ n ,~ measure of some compact convex R -subset
~+
is the vol n _ 1 valued concave and upper semi continuous Hence,
the
)] m
-
60 -
Rm.
(e.g.
I
n-I Gi (~I""'~m)'
=
0
(~I""'~ m )Erl., ~
if
elsewhere in
m
C.={xERm:K(X)nE.(x)n n D.(x)#¢)
where
~
~
j=1
and
]
is
G.
~
some
suitable continuous concave function on the compact
C ..
convex domain
~
Now our assumption on the vectors
clearly ensures that C.
~
which,
in turn,
aF
tions
---
Gi
~
vanishes on the boundary of
shows the continuity of the
Consequently,
aC
u.
the function
in
F
func(3.2)
is
~
Rm.
totally differentiable at any point of In what
follows,
K=Bn,
let
m=n+l,
and
let
the
be the normal vectors of the faces 1 ' · · · ,un T. of the given simplex TERn, directed outward from
vectors
u
T~
Denote by
E': ~
the
~
orthogonal
T
projection of
and define
(3.3)
face of
T*
supporting hyperplane of
(orthogonal) E~,
the
~
by
Ci. ~
we have for
p
0
Ci
Ci n
l
~
~
that
dI n1) --(vol [-TnB dp n p d --d p
=
+1
F(--, ... , - ) p p n+ 1 a.
-1
~
i=1
vol
p
n+ I
n n
{t: ( t, u
j = I
'>:0; ]
n+1
-
~
(i=I, . . . ,n+I).
Ci.=(U.,o':')
>
~
i=1
a. ---~-Ivol p
n+
n-
I[ Bnn{t:(
a. .....2} P
n-
t,u'> ~
=
~}n p
1
I[ PBnn{t:(
-
and by
0':
Let
~
ui ' .',
(the origin of
o
~
to
61
-
t,u'> ~
Ci .} ~
n
By
n+1
n n {t:(t.u'>:S ct,}] j=1
]
T':npB n
Observe now that ~
Ip2-ct~
R
• w1th
H . (0 1:)
~
O:S P
-
< Ict.l·
if
n-I
.~
~
~
=0
ER
n-i
(i=I ••••• n+l)
wise arbitrary). for the choice
= vol
n-
V
T.
~
2 + B n-I ] I[T.n (p 2 -ct.)
Hence. the case
~
~
of formula
c=o
= H. ( ~
T": ) ~
for H. :E. ~
~
-
(otherwe have
(i=I.2 ••••• n+I).
(3. I) follows with
lim vol [TnpB n ]. p"O n i. e ••
if where cone
T
n
is the
oET.
n-dimensional spatial angle of the
(O.oo)XT. But then the lemma also follows in the
stated generality. Viewing now the volume as a function of the radius we immediately have the following COROLLARY.
A (p) n
is
0': ~
(n-I)-dimensional affin sub-
in the
~
is a ball of center
~
= dist n (0.0":). and is p ~ Ict.1 ~ ~ Hence. considering an isometry
E~
space
E~npBn
where
~
and radius
here can be written in the
~
T~n(E~npBn)
form
]
d(n)
uously differentiable.
- 62 -
times contin-
PROOF.
For
continuity of some
k
~
2
t h at t h e f
n=l, AI (p),
is true.
, (d (k-I ) Hk I '
unct~ons
atives of order (~d(k»
and this
the claim is only the
the assertion holds for .
It follows
~.e.
d(n)=O,
-
,~
of
d(k-I)
then from
(
(3.1)
Suppose that for
i= I, . . . ,k+ I
)
,
.
der~v-
the
exist and continuous.
Ak _ 1 i ' ,
that
This means
k-I.
is
Ak
d(k-I)+I
times continuously differentiable over the set
(O,oo)'-.{al, .. ·,a k + I }· and to prove that
~
For ~s
Ak
~ ai'
Ak_l,i(
l '(~ 2 -a i2 ) + )
:= 0,
times continuously
d(k)
differentiable also in the points
it is
enough to show that
(3.4)
lim ~"a
But, by
(3. I) again,
constant
(i=I, . . . ,k+I).
°
. ~
ck_l,i'
Ak_l,i
if
h )
k-I ck_l,i
=
T
is small enough.
T
with some
Hence,
to show
(3.4) is the same thing as checking lim
(3.5)
(V=d(k)-I=[I] - I ) .
°
~" a
But
[~]-I 2
~
p.
j=1
where the follows.
]
(U
(~
k-l
2 -2- - j -a )
P.'-s are some polynomials, whence ]
(3.5)
By induction the Corollary is proved.
Taking the simplex
S
of Section 2 as an example
n
we see that the Corollary cannot, ed.
2
in general, be improv-
On the other hand, we saw that we have troubles with p's only, where the ball
differentiability in those
knocks against different dimensional
(n-I ,n-2, ••• ,n)
faces of the boundary of the
In fact,
easy to prove that
A
n
(p)
I-
simplex.
it is
is piecewise analytic on 63 -
(0,00) •
4. COMMENTS ON Because of
V (x) n
(2.2)
and Lemma 10,
V (x) n
has the
following form
V
n
(x)
n!A (p(x» n
(4. I)
n+1
n!pn(x>{S - ~ n
i=1
,
An- I , 3=··· =A n- I ,n+ I
where I
=--; A =A lIn n-I, I n - I =nn/2/ r
(I
2
+1), since
is easily seen that follows then, points of the
c
S
is an inner point of
n
sn
and
n S
It
n
do not have obtuse angles.
that the projections of
.
cn
(n-I)-dimensional faces,
will be inner
the projections
of these projections will be inner points of the -dimensional faces of the
It
(n-2)-
(n-I)-dimensional faces,
etc.
A .(/(~2-a~)+) by (4.1), the n-I ,~ ~ corresponding constants S I . will again be the So, when evaluating
n-
volume of the
,~
(n-I)-dimensional unit ball
and all the corresponding constants
i=1 , ... ,n+l,
a ..
(j=I, ... ,n) ~J will be positive, and this phenomenon is persistent with the decrease of dimension.
In this sense the recursion
in (4. I)
But one also notes that in
is "homogeneous".
the second step it will not be true that two of the
a's is the same and the rest is again the same
(i.e.
the ball reaches two faces at the same time, and, a bit later, it reaches the other faces again at the same time).
This "regularity"
disappears after the step, as
seen starting out from three dimensions.
-
64 -
Much work has been done to compile tables of percen2 wand similar statistics, in partic-
tage points for
n
ular by STEPHENS. in KNOTT [3J.
A survey and comparison can be found
In fact,
the most accurate.
Knott's results are proved to be
All these results,
on some kind of approximation of formula
Lemma
statistics, Knott.
Vn(x).
are based
In principle,
gives the possibility of the exact tab-
(4.1)
ulation.
tables,
10 is also applicable
e.g.,
M2n
for the
for other similar
statistic of Durbin and
seems to be accessible on a computer.
n=20
Unfortunately,
our computer facilities here are not
adequate at present to do this work. REFERENCES [I
J
S.
Csorg5, On an asymptotic expansion for the von Mises w2 statistic, Acta. Sci. Math. (Szeged), 38(1976),45-67.
[2J
H.
Hadwiger,
Vorlesungen
und Isoperimetrie,
Heidelberg, [3J
M.
Knott,
uber Inhalt,
Springer,
Berlin-Gottingen-
1957.
The distribution of the Cramer-von Mises
statistic for small sample sizes, J. Soc.,
S.
"-
Ser.
Oberflache
B.,
36(1974),
Royal Stat.
430-438.
Csorgo
and L.
Stacho
Bolyai Inst.
of Jozsef A.
Aradi vertanuk tere
I,
University
6722 Szeged, Hungary
-
65 -
,
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY).
1977.
NONNEGATIVE INFORMATION FUNCTIONS Z. DAR6CZY - GY. HAKSA
I.
INTRODUCTION
The notion of information functions has been introduced by Z. DAR6CZY [3J. A summary of the investigations concerning information functions can be found in the book [IJ by J.
ACZ~L and Z. DAR6CZY.
DEFINITION. A real-valued function the closed interval
and
f(O)
f
( I .2)
= f(I),
I f(-Z)
=
I
satisfies the functional equation x
f (x) + (I-x) f (-y-)
f(y) + (I-y) f(-I- ) -y
I-x
for all (1. 3)
is said to be an information
I]
if
function
(1. I)
[0,
defined on
f
(x,y)ED, where D
{(x,y): 0 S x
<
I,
0 S Y
<
I,
x+y S
I}.
The following result has been proved by Z. DAROCZY
,
and I. f
KATAI [4J.
Let
be an information function.
f
is nonnegative and bounded
f(x)=S(x)
for all
x E [ 0, I]
(above) where
-x log2 x -(I-x)log2(I-x) s(x)
( 1 .4)
=\
If
then
in [O,IJ
0
if
xE(O,I)
if
x=O
or
x=1
is the Shannon information function.
J. ACZEL and Z. DAR6CZY raised the following problem ([)J,
p.84):
S
Is
the only nonnegative information
function? Since the information function is the measure of information determined by the probability distribution
I-x}
{x,
(xE[ 0, I])
it is natural to suppose the
norlnegativity of this measure.
,
The result of Z. DAR6CZY
and I. KATAI has been generalized by G. DIDERRICH [5J in another direction: bounded in
[0, I]
If the information function
then
f
(x) =S (x)
for
f
is
x E [ 0, I] •
In this paper we prove two essential results which solve the problem of J. Aczel and Z. Daroczy. First, we prove that the lower hull of the nonnegative information functions is the Shannon information function, S (x)
for all
i.e.
= inf {f (x)
x E [ 0, I]
where
J+
denotes the class of non-
negative information functions.
Then we show that there
exist nonnegative information functions different from the Shannon information function,
that is
J
+"* {s}
.
Finally we give a negative answer to a question of P. FISCHER [6J. Throughout the paper denote the set of real numbers,
R, R+
and
Q+
the set of positive real
numbers and the set of positive rational numbers, respec-
-
68 -
tively. 2. THE LOWER HULL OF NONNEGATIVE INFORMATION FUNCTIONS We need the following result. THEOREM I.
The information function
f
is nonnegative i f and only i f there exists a ~
R+ -
R
such that
(2. I )
~(xy)
(2. 2)
~(x+y)
PROOF.
2
function
and
and
o~(X)+~(I-X)
if
xE(O,I)
if
x=O
It is known (se~ [IJ)
or
x=l.
that any information
function can be written in the form (2.3) where R
+
-
Since
I
~(2)
satisfies
R
=
I
2
and
(2.1)
for
~
:
x,yER+.
is nonnegative we have
f
f(x)
For
R
~(x)+~(y)
$
--I
f(x)
I
-
x~(y)+y~(x)
x,yER+
holds for
(2. 3)
I ~(2)
: [0,1]
(x E (0 , I) ) •
we have by
X,yER+
o
$
~( __ x_)+m(l_
which implies
x+y
't'
(2.2).
that the function
f
(2. I)
x) x+y
x+y
[~(x)+~(y)-~(x+y)]
,
Conversely it can easily be seen defined by (2.3)
I-
69 -
is a nonnegative
~
information function if
~
has the properties
I (-) 2
2'
(2.1) and (2.2).
LEMMA. Let
(2.4) where
J/.
n
be a natural number and
k S(2n)-
2n,k
S: [0,1]
-
R
2n
2n log2 (k
(k=O, I, ... ,2n)
)
is the Shannon information function.
Then the inequality
(2.5)
o
S J/.
is true for
< J/. 2n,k 2n,n k=O, I, .. . ,2n.
PROOF. For and
we have
k=0,1, .•. ,2n
J/.
2n,k
=J/.
2n,2n-k
thus it is enough to prove (2.5) for
J/.2n,0=0,
k=I,2, •.. ,n. We show that
increasing in
k
J/.2n,1
(k=I,2, ... ,n).
>
0
and
is
J/.2n,k
Indeed we have
2n-1 2n-1 J/.2n,1 = - -zn- log 2-zn- > O. On the other hand an easy calculation gives that for k=2,3, •.. ,n
J/.
2n,k
21n
-J/.
2n,k-1
f lOg 2(1+
2n-k
I
-log2(1+
2n-k)
k~l)
L
Since the sequence
I
(I+~)
i
~
is increasing,
This completes the proof of the Lemma.
-
70 -
k-I
1•
THEOREM 2. Let function.
(2.6)
f
be a nonnegative information
Then the inequality ~
f(x)
holds for all
S(x) xE[ 0, I] .
PROOF. By Theorem ~
: R
a(n)
- R + 1
= -n
is of the form (2.3) with
f
satisfying (2. I),
~(n)
(2.2) and
for every natural number
~(i) = n,
i.
Let
then (2. 1 )
implies a(nm)
a(n)+a(m)
for all natural numbers
n,m
and by (2.2) we have
(n+l)a(n+l) S na(n)
(n= 1 ,2, ••• ) •
Using a result of [4J (see also [IJ) we have (2.7)
This and the condition (2.8)
n=
(CER,
~
'21
1 (-) 2
~(n)
1,2" •.. ) . imply that
(n= 1 ,2, ••. ) •
Equation (2.1) gives that (2.9) for all
tER+
and
natural number and
k=I,2, • . . . Let now xE(O,I). By (2.1),
(2.9) we have
t
71
-
n
be a fixed
(2.2),
(2.8) and
0= cp(l) = cp(x+l-x)n]
n S
n k n-k CP(k)]X (I-x) +
~
k=O
k=O
n
~
+
n
~
n k n-k-I (k)X (n-k)(I-x) cp(l-x) +
k=O
-n 5 (x) + ncp(l-x) + ncp(x), n
where (2.10)
5
n
(x)
=
This means that for (2. II)
f(x)
=
cp (x)
and
xE(O,I)
+ cp (I-x)
~
5
n
degree function
where
2n
B 2n (X)
we have
(x).
- S2n(x) for is the Bernstein polynomial of
Let us consider the difference xE(O,I)
n"I,2, •••
B 2n (X)
of the (continuous) Shannon information 5:
[0,1] - R
(see [8]). We get
- '72 -
where
1 2n ,k
(k=O,I, .•• ,2n)
is the sequence defined
by (2.4). Applying our Lemma we have 2n
1
o S B 2n (x) -S2n (x) S 1- 2rl log2 ( n ) for
xE(O,I). The limit of the right hand side is
n ....oo
therefore lim s2n(x) Letting
n ....
=
lim B 2n (X)
=
sex).
n ..... oo
n-+o;,
00
in (2.11) we get the
st~tement
of
Theorem 2. 3. NONNEGATIVE NON-SHANNON INFORMATION FUNCTIONS We shall use the following result THEOREM 3. There exists a function identically zero such that (3. I)
d(xy)
xd(y)+yd(x)
and (3.2)
d(x+y)
d(x)+d(y)
\
- 73 -
(see [9]). d
R -
R
not
are satisfied for all
x,yER.
REMARK. A function (3.2) for
x,yER
d:
R -
R
satisfying (3.1),
is called a derivation on the field
R.
THEOREM 4.
There exists a nonnegative information
function different from the Shannon information function.
PROOF. Let
d:
derivation. Then
d
R -
be a non identically zero
R
is not identically zero in (0,1)
therefore the function d 2 (x) x(l-x)
if
xE(O,I)
if
x=O
f(x)
(3.3)
is nonnegative in [0,1] information function. I
points we have
f(I)
= I
is zero at rational
d
f(O)
and
f
= f(I). defined by (3.3)
satisfies the functional equation (1.2) (x,y)ED.
x=1
and different from the Shannon
Since
We show that the function Every derivation
d 2 (x)=d 2 (I-x),
or
d
for all
satisfies the identity
thus it is enough to show that the func-
tion A(X,y)
is symmetric for
d 2 (x) x(l-x)
°< x
+ (I-x)
< I,
d 2 (-y-) I-x -y_(I- -y-) I-x I-x
°< y
some calculation we get
-
74 -
< I,
x+y
< I. After
I-y d2(x) + I-x d2(y) + x(l-x-y) y(l-x-y)
A(x,y)
+ 2 d(x)d(y)
I-x-y which proves the symmetry of
and Theorem 3.
A
4. REMARKS (0
Let
d
R -
be a nonidentically zero deri-
R
vat ion and d 2 (x)
<\>(x)
(4. I )
The function
x
<\>
R+ -
(4.2)
<\>(xy)
(4.3)
<\>(x+y)
s <\>(x)
(4.4)
<\>(r)
°
and
<\>
- d[d(x)] has the properties
R
x<\>(y) + y<\>(x) +
(x, yER+),
<\>(y) '-
is not a derivation.
This shows that the function
~
defined by
(4.5)
satisfies the assumptions of Theorem I, therefore
£(x)
(4.6)
_I :(X)
if
+ ,(I-x)
if
xE(O, I)
x=O
is a nonnegative information function.
\
-
75 -
or
x=1
After an easy
calculation we have
f(x)
-x
- d[ d(x)] -d[ d(l-x)] for
xE(O,I), i.e.
d 2 (X)
S(x)
+ x(l-x)
the function defined by (4.6)
is
(3.3).
identical to the nonnegative information function
(ii) Our results make it possible to answer a problem of P. FISCHER [6J in the negative MOSZNER [7J, and E. a function
X : R+ -
(4.7)
X(x)
(4.8)
x(rx)
(4.9)
X(x+y)
and
(4.10)
where
with the properties (xER+),
0
rx(x)
(xER+, rEQ +),
::; X(x)+X(y)
(x,yER+)
II T(x) II
x(x) T
: R
normed linear (4.11)
R
cannot be written in the form
X
where
~
(see also Z.
BERZ [2J). We prove that there exists
-
E
is an additive function and
~pace.
d: R -
R
(xER+)
is a derivation not identically zero.
(4.7) and (4.8) are obviously satisfied by this Further for all
is a
Let namely
d 2 (x) x
X(x)
E
x,yER+
the inequality
-
76 -
X.
2 2 2 2 x d (y)+y d (x)-
o S [xd(y)-yd(x)] 2
-2xyd(x)d(y) or xy[ d holds.
2
This means that x(x+y)
i.e.
2
(x) +d (y) +2d (x) d (y)]
(4.9)
d 2 (x+y) x+y
is valid. Assuming
o
+
x
d 2 (y) y
=
X(x)+X(y),
(4.10) we would have (n=I,2, ••. )
IIT(n)D
hence d 2 (x) x+n
d 2 (x+n) x+n
would follow for all
HT(x+n)O
=
IT(x)ft
xER+, n=I,2, . . . . Letting
n -
we get IT(x)H
o
which contradicts the fact that
d
is
a derivation
which is not identically zero. REFERENCES [I]
J. Aczel-Z. Daroczy, On measures of information and their characterizations, Academic Press, New York,
1975.
\
-
77 -
00
[2J
E.
Berz, Sublinear functions on
Math.,
[3J
Z. Daroczy, On the Shannon measure of information (in Hungarian), Magyar Kozl.,
[4J
R, Aequationes
12.(1975), 200-206.
Tud.
Akad. Mat.
Fiz.
Oszt.
19(1969),9-24.
Z. Daroczy -
I. Katai, Additive Zahlentheoretische
Funktionen und das Mass der Information, Ann. Univ. Sci.
[5J
Budapest Eotvos,
Sect. Math.,
13(1970), 83-88.
G. Diderrich, The role of boundedness in characterizing Shannon entropy, Information and Control,
[6J
149-161.
P. Fischer, Remarque 5 - Probleme 23, Aequationes Math.,
[7J
29(1975),
1(1968),300.
Z. Moszner, Sur une hypothese au sujet des funct·ions subadditives, Aequationes Math., 2(1969),
380-
-386. [8J
I.P. Natanson, Theorie der Funktionen einer reellen Veranderlichen, Akademie Verlag,
[9J
O. Zariski and P.
1954.
Samuel, Commutative algebra, D.
Van Nostrand, Princeton,
1958.
Z. Daroczy and Gy. Maksa Department of Mathematics University of L. Kossuth 4010 Debrecen, Pf.
Berlin,
12
Hungary
-
78 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
RECURSIVELY DEFINED MARKOV PROCESSES (DISCRETE PARAMETER) W. EBERL
I.
INTRODUCTION
Let
(X) n
be a MARKOV process defined over
(~,A,p)
by a recurrence relation ( 1)
X
n+
1(')
= X
n
(.)+.1 R
n n
R
n
n
(.»+G
Xo = -xER k ,
with an initial condition (2)
(X
n+
I(X ( . ) , . )
n
(nEN O )
where
R+r :R k n
(nEN) ,
(3) 00
(4 )
a
n
> 0
a
n
A great number of procedures considered in the theory of stochastic approximation can be represented by a recurrence relation (I) with conditions (2)-(4).
\
-
79 -
They deal with the problem of the almost sure convergence of
on the set
(X n )
{XER
k
IR(x)=O}.
In this connection various statements were made in a series of papers mainly especially for Robbins-Monro or Kiefer-Wolfowitz procedures. For
R
having only one
zero, e.g. ROBBINS-MONRO [9J, KIEFER-WOLFOWITZ [6J and GLADYSHEV [5J have shown the almost sure convergence to this zero point under certain conditions. For the case
when
R
has finitely or countably many zeroes,
e.g. FABIAN [3J,[4J, KRASULINA [7J and NEVELSON [8J have proved the almost sure convergence to the set of the zeroes resp.
to zeroes.
If
is allowed to have
R
arbitrarily many zeroes finally,
BRAVERMAN-ROZONOER [IJ
reached the almost sure convergence to the set of the zeroes. But from this last result of BRAVERMAN-ROZONOER in the general case we have no further information about this convergence. Our aim will be in the sequel to get more knowledge about it. Before introducing some notations and giving a lemma we formulate a sufficient conditions for a process
defined by (1)-(4) to
(x )
n
be Markovian. REMARK I. It can be readily verified that the process
(x) n
defined by the recurrence relations (I)
is Markovian for given functions numbers
a
n sing sequence
R,rn,G n and real with (2)-(4) if there exists a nondecrea-
(A ) n
of sub-a-algebras in
Bk -measurable
(a) (b)
(Gn+l(x,.»
k
and
A
xER
-
80 -
n
A
such that
(nEN) ;
are independent
(nEN).
2. NOTATIONS AND A LEMMA For any function
R+
is measurable for all
LV(n,x)
(5 )
R+,
E(V(n+I,X
V (B) E d(x,B)
with
4>(B)
(7)
{xld(x,B)
inf yEB
Ixn =x)-V(n,x)
E}(){xllxl
?:
k
inf n?:n O
xEV
LEMMA I. Let (X)
>
cp(n,x)
O}.
(B)
c
(2)-(4),
BCR
k
formula
and let there exist
V:NOXRk
R+
and a
such that
cpE4>(B)
~
LV(n,x)
(a)
:
be defined by recursion
a radially unbounded* function function
0)
following lemma holds.
n
with conditions
(1)
>
(E
E
> O:3nOEN
R+lvE
With these notations the
< ..!..}
and
Ix-yl
{cp :NOXR
I)
nEN O ' we put
BCR k
Further let us define for
(6 )
n+
V(n,.):
for which
-a cp(n,x) n
Then :.':
We call a function
V(n,') a.
>
0
: Rk
-+
VENOXRk
s>0
there exists a
imply V(n,x)
-+
R+
radially unbounded iff
R+ is measurable for all
>
such that
a.. -
\
81
-
nENo
Ix I >
and for every Sand
nEN
o
(i)
P{sup!xnl
(ii)
P { 1:
<
co}
=
I,
co a
n
n=O (iii)
cp(n,X ) n
P {lim d (x ,B)
<
co}
O}
n
I,
I.
PROOF. From the MARKOV property and the condition (a) we conclude
E ( V ( n + I ,X
n+
I
I
I ) X O ' • • • ,X ) = E ( V ( n + I ,X I) X ) n n+ n ~
Therefore
~
a.s.
V(n,X)
n
constitutes a nonnegative super-
(V(n,X»
n
martingale, which according to a well known theorem (see [2J)
converges almost surely to a finite limit. This
fact and the assumed radial unboundness of
V
imply now
the conclusion (i). From (5) we obtain
(8)
LV(n,X) n
=
E(V(n+I,X
n+
1)lx )-V(n,X) n
Taking expectations on both sides of the first
(9)
j=O
(8) and then adding
equalities we get
n+1
E[ ~
n
LV(j,Xo)] ]
The validity of
E(v(n+l,xn+I»-V(O,X O )
=
(nENO).
(9) leads us together with the first
statement of this lemma to
E[ ~
j=O
a oCP(j,Xo)] ]
]
~
V(O,X O )
and therefore to (ii). To prove (iii) finally, we can
-
82 -
infer from (4) and from (ii) that there exist functions T
n
:
n
NO
such that
o
lim cp(Tn,X T ) n-- oo n
a.s. cpE~(B)
Because of the assumption
and because of
(i)
this implies the third statement. 3. A CONVERGENCE THEOREM We are ready now to formulate our convergence the orem.
(x)
THEOREM I. Let relation
with conditions
(1)
B = {xERk!R(X)=O} tion
V: NOXR
(a)
be defined by recurrence
n
k
(2)-(4),
let
and exist a radially unbounded func-
R+
and a
function
cpE~(B)
with
LV(n,x) S -a cp(n,x) n
Further let the following conditions hold: 00
<
(b)
l: a b n n n=O
(c)
sup IR n (x) nENO
sup I r (x) n xERk
00
with
b
I
= R(y}
<
00
l: Gn+I(Xn(·),·)1 n=O
<
00
n
I,
(yER+) ,
IxlSy I
(d)
Then
(X ) n B
closure
a.s.
converges almost surely to a point of the of
B
or to the boundary
\
-
83 -
oB
of
B.
PROOF. We proceed in three steps, assume that
(d) holds for all (I) For all
nO=nO(w)ENO X (w) n
w.l.o.g.
wEn.
for which there exists an index
wEn,
X (w)EB
such that
(n~nO)'
n
the limit of
exists. Indeed, we obtain for such an Ix (w)-X (w)I=!
n
m
n-l
the inequality
wEn
n-l ~ [a.r.(X.(w»+G. l(X.(w),w)] . ] ] ] J+ ]
J=m
n-l
~ a.b.+1 ~ G. l(x.(w),w)1 j=m ] ] j=m J+ ]
$
with the aid of assumptions
(b) and (d) this inequality lim X (w).
implies the conjectured existence of
n
(II) Next we show that (10)
=
P{lim d(X ,B) n
O}
I.
For this purpose, let us choose an
according
wEn
to the Lemma, namely such that (I I)
sup n
(12)
EN
Ix (w)1 n
I:
a
0
n=O (13)
n
e=e:(w)
>
00,
< 00,
(w»
n
numbers with (14)
~(n,X
lim d(X (w),B)
I f we assume
an
n
<
y
=
O.
further
lim d(X (w),B) n and sequences (m .) ,
0
]
m. ]
<
< mj
n. ]
X
n.
(jEN)
+1
(w )EB
]
-
e
,
84 -
>
0, we can select
(n j)
of natural
such that
X (W)$B n e
(m.$n
]
$
hold for all (I),
JEN, where
B£
< £}. Using
{XERkld(x,B)
(14) and assumption (c) we get the inequality
(I I),
n .-1
o
<
£
~
Ix
n.
(w)-x
J
(w)1
m.
~
R (!/)
J
~
a
n=m.
J
+
n
J
n .-1 +
I
J
~ Gn + 1 (Xn(w),w) n=m.
I
(jEN) •
J
is large enough, because of the assumption (d)
If
we have n .-1
o <
( 1 5)
£
<
J
2R (!/)
a
~
n=m.
n
J
~E~(B)
The condition j2EN
and
0
o < 0
(16)
>
and (I I) ensure the existence of
such that
0
~
~(n,X
inf m .~n
J
J
.
n
(w»
MUltiplying inequalities (IS) and (16) we obtain for all n .-1 J < O£ < 2R(Y) ~ a
o
n=m.
n
~(n,X
n
(w»
J
for all
j
(III)
~ max(jl,j2)'
in contradiction to (12).
A ={xERkld(x,B C )
With
£
verify now for almost all existence of (n
~ nO)
or
wEn
nO=nO(w.£)ENO
Xn (w)EA 2 £
(n
<
£}
>
(£
and for all
such that either ~ nO)
we shall
0) £
>
0 the X (w)4A n
£
holds.
To this end assume the contrary that we can choose of natural wEn, £ > 0 and two sequences (m j) , (n j) such that numbers with m. < n. < mj + 1 (jEN) J
(17)
J
X
(w)EA, X (w)EB
n\ _.: _n
(m. J
~
n
<
n .) J
holds for all JEN. By the aid of (I), (17) and of the assumption (b) we are led to the inequality n
<
o
~
£
Ix
n.
(w)-x
]
(w)1
m.
~
]
j
-I
I: a b + n n n=m. ]
n .-1 ]
+
(jEN) •
I: G n+ I(X n (w).w)1 n=m. ]
But in view of the assumptions inequality contradicts
(b)
and
(d),
this last
> O.
£
From (1)-(111) we deduce immediately the assertion of Theorem I. As a matter of fact,
wE~
if for
there
exist and therefore it follows the existence
X (w)EB n
lim X (w). which is obviously an element of
of
n
the other hand,
£0
> 0
and
for almost all
nOEN
wE~
without
>
£
there exists an
0
On
such an
it follows according to step
the proof that for all
13.
(III) of nOEN
X n (w)EA 2 £ (n ~ nO). Combining this with part (I) of the Proof finally we obtain the assertion of Theorem
with I •
Condition (d) of the just proved theorem is somewhat disagreeable, since its (direct) verification usually requires the knowledge of the paths of
I (x
(G
(.),.)),
n+ n whereas in view of the theory of stability (Ljapunov's
direct method) one is interested in convergence conjectures with respect to on
(x ). n
(x ) n
without special assumptions
Under the condition of Remark
I
for
(x) n
to
be Markovian, now we give sufficient conditions for the validity of assumption Cd)
in Theorem I.
REMARK 2. Under the condition of Remark 1 the assu.ption
(d)
of Theorem 1 is fulfilled,
-
86 -
if
(a)
E(G
n
o
(x,.»
(nEN,
k
xER ),
(b)
holds.
Indeed, n
in this case it can be shown that
~ G +1 (x (.),.) constitutes a martingale with j=O n n bounded second moments and hence it converges a.s. to a
Z (.) = n
finite limit. We omit a detailed proof. Nevertheless, condition (b) of the preceding remark still has the disadvantage to depend (explicitely) on the recursively defined
X. n
The following remark shows how
to get rid of this handicap. REMARK 3. Under the condition of Remark 1 the assumption
(d)
of Theorem 1 is satisfied, i f
(a)
E(G
(b)
l/>n+ 1 (x)
n
(x,.»
(nEN,
= 0
E(IGn+l(x,.)1
2
)
XER k ),
S gn(I+(V(n,x» k
«n,x)ENOXR ) 00
holds, where
~
g
n=O
n
To prove this,
<
00.
first we notice that for all
the random variables
and
X
n
nENO
are
independent under the condition (b) of Remark t; hence we have k
(nEN O ' xER )
and therefore with regard to condition (b) of Remark 3
\
- 87 -
( I 8)
g
n
(I+E(v(n,x )))
n
From the proof of Lemma I we know that
(V(n,X
n
is a
))
supermartingale, and so the sequence of expectations is ~
E(V(n,X )) n
bounded:
C. Then we obtain from (18)
thus we have proved condition (b) of Remark 2 and at the same time condition (d) of THEOREM I as well. 4. COROLLARIES Finally, we formulate and prove some corollaries of Theorem I which are special cases of this theorem under appropriate additional assumptions. COROLLARY I. If in addition to the assumptions of lim a
Theorem 1
n
0
=
holds and furthermore
union of finitely many pairwise disjoint, B.
~
(I
~
~
i
n),
converges a.s.
(X) n
or to one of the boundaries
dB i
(I
B
is the
closed sets
to a point of ~
i
~
B
n).
PROOF. Under the above assumptions we deduce without difficulties that lim As we know
Ixn+ I -xn I
=
a. s.
0
min d(B.,B.) > 0, this implies the statement
I~i<j~n
~
]
of Corollary I as an application of Theorem I.
-
88 -
COROLLARY Z. Corollary 1 remains valid for the case of
B
being the union of countably many pairwise
disjoint,
closed sets
inf
B.
>
d(B.,B.) ]
~
I~i<j
i f we suppose
(iEN),
~
O.
The proof is obvious. COROLLARY 3 (NEVELSON [8J).In addition to the assumptions of Theorem 1 let
R
be continuous,
consist of countably many elements and let Then
(x )
n a zer'o of
converges a.s.
to a point of
B
lim a B,
O.
n
i.e.
to
R.
Choose an wE~ such that suplx (w)1 = n lim d(X (w),B) = 0 and limlX I(w)-x (w)1 n n+ n 0; with regard to statement (i) of Lemma I, step (II) PROOF.
y
< 00,
of the proof of Theorem I and the proof of Corollary these conditions hold for almost all
wE~.
Hence the (x (w»
corollary is proved, if we succeed in showing to have one limit point only. To this end, a limit point of point with
(x
n
(w»
and let
Xz
n
let
be
be an arbitrary
xl#x Z • On account to the assumptions
is an element of
B.
As
continuous there exist
B
is countable and
R
is
<
Z£
< {)
and
{)
xI
£
with
0
such that (19)
and such that not
K{),£(X I )
{x!{)-Z£ ~ lXI-xi ~ {)+ZE}
=
contain any element of
B.
lim d(X (w),B) n that
\
-
89 -
As a result of
does
d(X
(20)
n
< e:
(w),B)
on the other hand, according to there exists
selecting now
X
n
(x
n
Ko,e:(X I ) does not
we obtain
B
(w)EU~
u-e: (xI)
Hence, with regard to (19) of
such that
(21) and because
contain any element of
o
e:
n3 ~ max {n I ,n 2 }
in view of (20) and
Ix n+ I (w)-X n (w) I
such that
n 2 =n 2 (w,e:)EN
Ix n+ I(w)-X n (w)1 <
(2 I)
lim
cannot be a limit point
X2
(w».
REFERENCES [I]
E.M.
Braverman - L.I.
Rozonoer,
Convergence of random
processes in the theory of learning machines, I, II, Avtomatika i
3(1969), 55-77, 87-103.
Telemehanika,
[2]
J.L. Doob, Stochastic processes, Woley,
[3]
V.
Fab~an,
Stochastic approximation methods, Czehos-
lovak Math.
[4]
V.
1953.
J.,
10(1960),
123-158.
Fabian, A new one-dimensional stochastic approx-
imation method for finding a local minimum of a function,
Transactions of the Third Prague Conference
on Information Theory,
Statistical Decision functions,
Random Processes, Prague,
-
1964, 85-105.
90 -
[5J
E.G. Gladyshev, On stochastic approximation, Theory of Probability and its Applications,
10(1965),
275-278. [6J
J. Kiefer - J. Wolfowitz, Stochastic estimation of the maximum of a regression function, Ann. Math. Statist.,
[7J
23(1952), 462-466.
T.P. Krasulina, On the Robbins-Monro procedure for the case of several roots, Theory of Probability 12(1967), 386-390.
and its Applications,
[8J
M.B. Nevelson, Convergence of continuous and discrete Robbins-Monro procedures in the case of a multiple-root regression equation; Problemy Peredaci Informacii, 8(1972), 48-57.
[9J
H. Robbins -
S. Monro, A stochastic approximation
method, Ann. Math. Statist.,
22(1951), 400-407.
W. Eberl Fernuniversitat Hagen Fachbereich Mathematik LUtzowstrasse 125, 5800 Hagen, GFR
\
-
91
-
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION IffiTHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
DISTRIBUTION OF THE SUM AND THE MEAN OF MIXED RANDOM VARIABLES IN A CLASS OF DISTRIBUTIONS T. GERSTENKORN
INTRODUCTION Research work connected with mixed distributions has a rather rich literature and the interest in the problews goes back to the end of the last century. The main point of the mentioned investigations was to give estimators of the parameters of mixed distributions. The present paper does not yield a bibliography concerning the question, since the problem, treated here, is of a different nature. Namely, our aim is to establish the form of the probability distribution for both the discrete and continuous cases for a random variable which is either the sum or the mean of mixed random variables. Section I includes the conditions, given by the form and properties of the characteristic function, sufficient for the distribution to be a mixture of the same type as that of the components, however, with some change in the weights of the mixture. Thus conditions for the validity of the so-called "addition theorem" for mixed distributions are formulated there. The /
93 -
distribution of the mean can be obtained as a corollary from the basic result concerning the sum. Section 2 indicates the possibility of specific applications of the results given in Section I. I. MIXED DISTRIBUTIONS
Let the random variables
(1-1,2, •.. ,r)
xl
be of
discrete or continuous type with probability functions is the probability of the event
in the discrete case, and it is the density
Xl-x
function in the continuous case. A1 set, i.e.
A1-{all,a21, •.• ,as1}
(s ~ I
which the distribution depends. Let r
positive numbers such that
~
1-1
is the parameter
a -I 1
al
integer), on be arbitrary
(r ~
2).
DEFINITION. We shall say that the random variable
X
has a mixed distribution if its probability function
has the form r
(I)
~
P(x,A)
1-1 where
A
a l P l (x,A 1 )
is the parameter set of the mixed distribution.
We have the following LEMMA.
The characteristic function of the mixed
distribution has the form
(2)
~(t,A)
~
r ~
a1~1(t,A1)
1=1 where
~1(t,Al)
random variable
is the characteristic function of the Xl.
Using the notation
-
94 -
we have the following THEOREM. Let the random variables X k
(k=1 ,2, ... ,n)
be independent and have identical mixed distributions defined by (1). ponent of
Xk
Let the variable
be the
Xk~
£-th com-
with characteristic function h(t,AQ,)
(3)
g(t) is an additive function of the parameter
A~.
Then the variable
Y = -r,X k has a mixed distribution of the same kind, but with new parameters and weights: k~ r
P(y,A)
(4 )
n --, k n •
-r,
n!
a.~
k 1+ •• • +k r =n ~= 1
p(y,
'"
PROOF. Assuming that the characteristic functions of Xk £
the random variables
have the form (3) and applying
the lemma we obtain the characteristic function of the variable
Xk
with mixed distribution in the form h(t,A~)
r
(5 )
(t,A)
CPX
-r,
a.~g(t)
~=I
k
Taking into account the independence of the variables formula
Xk '
the coincidence of their distributions and
(5), we find that the characteristic function of
the random variable
Y
is of the form
(6 )
Due to the multinomial
Newton formula and the required
- 95 -
additivity with respect to the parameter k r
n!
~
(Xi
h(t,
we get
r
i
r k.A.)=h(t, ~ k.a l ., j=1 J J j=1 J J ~
~
h(t,
j=1
IT - k' g(t) i· k l +·· .+kn =n i=1
r
where
Ai
k.A.)
J
J
r
r k.a 2 ., ••• , ~ k.a .), j=1 J J j=1 J sJ ~
r
i. e. A = {
~
k.A .,
j= 1 J
J
Considering the fact that the probability distribution is uniquely determined by the characteristic function, we obtain (4). As a corollary we can obtain the
x
expression for
=
n
corres~Jnding
Y.
2. APPLICATIONS Four examples related to often used probability distributions will be given for the application of our theorem. a) The Poisson distribution. The characteristic function of the Poisson distribution is exp{-"[ I-exp(it)]}. By taking
g(t)=e, h(t,Ai)=-"i(l-exp(it»
we have
possibility to represent the chracteristic function of the variable
Xi
(a component of the mixture) in the
form (3). Taking into account all further assumptions and using the additivity of the function to the parameter
"
h
with respect
we can easily show that the
distribution of the mean of the variables with mixed Poisson distribution is of the form (4), with parameter
- 96 -
b) The normal distribution. The characteristic function of the normal distribution can be represented in form (3), under the assumption that (to" ~)
g(t)=e,
h(t,A~)=
2
2
. The function
h(t,A~)
is
additive with respect to both parameters which occur here. Thus in the case of the normal distribution we have
(4), where
P(x,A)
is the density of this distribu-
tion with parameter
c) The Cauchy distribution. For the random variable x~
the Cauchy probability law, with parameters
A~
(-00
<
~~
<
+00,
A~
>
~~
and
0) specified by the probability
density function
has the characteristic
That is,
function
g(t)=e, h(t)=it~~-A~ltl,
the latter function
being additive with respect to both parameters appearing r
in it. Thus we have (4) , where
A
=
r
{ ~ k~]J~, ~=I
~
~ k~A~}. =1
d) The binomial distribution. In this case we have it g(t)=I+p(e -I) and h(t,A~)=const. It is evident that the addition theorem holds here with respect to the size of the sample. -
97 -
The distributions of the mean in the Poisson and normal cases have already been given by means of a special procedure in the paper quoted below. REFERENCE [I]
H.
Jakuszenkow,
Rozk~ad
kiadow, Zeszyty Nauk.
~redniej
w mieszaninie roz-
Politechnika ~odzkJ,
(1973), 231-237.
T. Gerstenkorn Mathematical Institute University of Lodz 90-238 Lodz, Poland
- 98 -
3, Nol68
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
LOCALLY MOST POWERFUL AND OTHER TESTS FOR INDEPENDENCE IN MULTI-VARIATE POPULATIONS
Z. GOVINDARAJULU - A.P. GORE
SUMtfARY
I.
This paper is concerned with the problem of testing the hypotheses of complete independence in a
p-variate
population P
( 1. I)
where
HO : F(XI,···,x p ) F
and
joint and the
F.
~
IT i-I
(i-I, ... ,p)
~arginal
F. (x .) ~
~
are respectively the
distribution functions,
assumed to be continuous.
all
In the parametric set up, the
test statistic used is a linear combination of all sample bivariate product-moment correlation coefficients. In the nonparametric set up, a statistic analogous to the above is a similar linear combination 0f all sample bivariate rank correlation coefficients, while two others are based on certain multivariate analogues of bivariate rank correlation. All the tests reject the null hypothesis whenever the corresponding test statistic is too large. Asymptotic distribution theory for the
- 99 -
test statistics is developed and asymptotic relative efficiencies of the various tests are studied. Finally, a locally most powerful test for a class of mixture alternatives is derived and its asymptotic properties, especially, that of the normal scores test for the -variate case, are studied.
p-
In particular, the local
optimality of some of the statistics, considered earlier, is established. 2.
INTRODUCTION AND THE VARIOUS TEST CRITERIA
Tests of independence in
~ultivariate
populations
have been proposed in the literature under parametric as well as nonparametric assumptions. ANDERSON [IJ has given a likelihood ratio-test criterion for the case of multivariate normal distributions.
CONSUL [6J has worked
out the related distribution theory in detail. [15J
LAWLEY
(see also MORRISON [17J) has suggested the use of a
statistic proportional to sum of squares of all bivariate product moment correlations as an approximation to the likelihood ratio test. HOEFFDING [I IJ has given a nonparametric test of independence for bivariate populations. He has also shown that there do not exist tests of independence based on ranks which are unbiased for any significance level with respect to all continuous alternatives. However, LEHMANN [16J has demonstrated that the one-sided test procedures based on Kendall's tau and Spearman's rho are unbiased for all continuous
(X,Y)
such that
sion dependent on
X
is positively (negatively) regresX,
for given
that is, the conditional distribu-
tion of
Y
ing) in
x. He has also pointed out that positive
X=x,
is nonincreasing (nondecreas-
(negative) regression dependence implies positive (negative) quadrant dependence, -
100 -
that is, covariance of
X,Y
is nonnegative
(nonpositive).
BLUM, KIEFER, and
ROSENBLATT [3J have proposed Kolmogorov type and Cramer-Von Mises type tests.
BELL and SHITH [2J have cha-
racterized all distribution-free tests for independence, and for the case of total independence they have characterized the family of all rank tests. The test proposed here are in the spirit of Kendall's tau except for the one based on product-moment correlations. Theorem 5.]
In
is derived a locally most powerful test
against a class of alternatives and which, in particular, includes one of the tests proposed in Section 2. As I
GOKHALE []OJ has pointed out, dence,
in testing for indepen-
the knowledge of alternatives
by the practical problem at hand)
(possibly suggested
is extremely important.
We shall consider a subclass of alternatives in which either all the bivariate correlations or their algebraic signs are known. This seems to be reasonable, because in a given situation one would know whether a pair of variables, are more likely to increase or decrease together.
The tests are consistent against appropriate
alternatives. Any functional of the tau type does not characterize independence, in the sense of reducing to zero,
if and only if,
the variables are independent, and
the statistics used here are designed to test whether the relevant functionals are zero.
Consequently, for the
subclass of alternatives for which the functional in question is zero, power of the test reduces to its level of significance.
This characteristic is also inherent in
the test proposed by PURl, SEN and GOKHALE
[]9J for the
same problem. They define nonparametric association matrix whose elements are measures of association H O' a in terms of the elements
between two variates. For each null hypothesis modified null hypothesis
H*
o
101
of the association matrix is set up and tested. For instance, if and
8"
measures the association between
~J
j-th variates and
of the variates, q
is that first of the other
H~
is
for all
8 ij =0
variates
(out of
variates, H~
p-q
i-th
asserts pairwise independence
HO
p)
i¢j. Or if
are independent
becomes
•.• ,q; j=q+I, •.. ,p). In all cases
HO
8, ,=0 ~J
(i=I, ...
implies
H* but 0 Thus the procedures
H
o
the converse is not necessarily true.
given by PURl et al [I9J are appropriate for a subclass of the class of all alternatives. Further, their tests constitute nonparametric analogues of the parametric likelihood ratio tests. The approach of the present paper is similar to Puri et al regarding tests of
T2
& T3
TI
and dissimilar in case
T 4 . Heuristically, the latter test proced-
and
ures are directed to the problem of complete independence, and are based on
p-variate measures of associa-
tion, rather than a collection of bivariate measures. They may be looked upon as generalizations of Kendall's tau.
It is also of interest to note that
and
are equivalent. X~
Let
-~
(i=I,oo.,n)
= (XI"oo.,X ,) ~
p~
F(X I , ... ,x p )' Let
random sample from
denote a
sjk
denote the
sample product-moment correlation coefficient between j-th and
k-th variates
(I
S
S p;
j, k
j
¢
k).
Let
P jk
denote the corresponding population parameter. Let
e jk
be some given constants, which, in particular are:
C jk= I
~f 4
> 0
P jk
and
C jk =-1
~f 4
the statistic (2. I )
T
I
=
p
p
~
~
cjksjk
j=I k=1 p(p-I) j¢k to test the null hypothesis (1.1).
-
102 -
P jk
< 0.
We propose
Two vectors of the same dimension are said to be concordant if all the differences between corresponding co-ordinates of the two vectors have the same sign. Define
x-£
if
.c'.m)
(2.2)
(2.3)
T2
0
= /
n
n
~
~
and
X -m
are concordant,
otherwise, ¢(£,m) n (n - 1 )
£=1 m=1 £*m
Let
Tjk
n
n
~
~
2¢ j k (£ , m) - 1 n(n-I)
£=1 m=1 £*m
where
(2.4)
"jk(,·m)
if
=/'
T
=
p
P
~
~
(x
°
Jm
'Xk
m
)
are concordant
o
(2.5)
and
otherwise,
CokT Ok J
j = 1 k= 1 jH
J
P (p-I )
and
(2.6)
p
p
~
~
j=1 k=1 jH
Also,
let
(which,
a
°
~
C
J°kT JOk
P (p- 1 )
(i=I, ••• ,p)
in particular d
be known bounded constants
can take
-
103 -
+1
or
-I)
such that
n
(
np n-
1;(
p-
I)
p ~
a
i=1
t
<
~ [2 i=1 ~ a.I(x. -x. )m=1 ~ ~~ ~m
9..l-m
(2.7)
where
p
n
~ ~=I
I(t)
.J
2
~
equals
if
~
t
0
and equals zero if
O. I t is shown in Section 3 that
related to T3
is linearly
and hence constitutes an equivalent way
of formulating a multivariate rank correlation. T4 essentially attempts to improve upon use all the information in the data.
which does not
T2
This is readily
apparent from the fact that a pair of vectors in which even one difference in the corresponding coordinates goes in the direction opposite to the other
p-I diff-
erences, is classified as discordant.
3. ASYMPTOTIC DISTRIBUTION THEORY 3.1.
Distribution of
(3. 1 )
p
p
~
~
j=1 k=1
T 1 • It can be shown that
CjkP jk p(p_l)
J
j I-k
is asymptotically normally distributed provided that the Xj~'S
under
have finite fourth moments, the asymptotic variance HO
being
3.2. Distribution of
statistic with In(T 2 -Tl c )
variance
~(~,m)
T2 •
Clearly
T2
is a
u-
as its symmetry kernel and
is asymptotically normal with zero mean and as an immediate extension of a -
104 -
result
in NO ETHER (CI8J. n
(3.2)
n
When
=
c
is
70) where
EmU-.m).
cc = P (XI -
HO
p.
~2
is concordant with
true the variance of
~3)'
and
reduces to
T2
8[ 2 P (2 P +I)-2'3 P ] /(12)P.
Distribut~on
3.3.
(see
(2.4)
and
n
I ~ n(n-I).Q.=I
(3.3)
of
T3 •
n
I P P(P_I)j:1
~
m=1
.Q.oFm
Thus
it is a
can be rewritten as
T3
(2.6)) P
k:ICjk(2~jk(t,m)-I).
jH
U-statistic with the expression in the
square brackets as its symmetric kernel. application of Theorem 7.1 [9J.Sec.
5.5 and Thm.
Hence as an
of Hoeffding [12J
5.1, pp.
224-225)
(see Fraser is
In(T 3 -T)
~asymptoticallY normally distributed with zero mean and variance
4[Ep
-2
(p-I)
(3.4)
-2
(~~ C)'k{2~)'k(t,m)-I})x joFk
8
which reduces to 9p 3.4.
(3.5)
2
(p-I)
2
Distribution of
p(p-l)n(n-I)T 4
2 ~ ~ C)'k
holds.
Notice that
T4 •
n
n
~
~
.Q.=I m=1 .Q.oFm
-
when
jH
105 -
[4{I: a.I(x.n-x.)} ~
~~
~m
2
+
(~a.)2-4(~a.)(~a.I(X'n-X. )]. 1 ~ ~ i ~ ~~ ~m
+
which further simplifies to n n p 2 ~ ~ [4 ~ a.I(x.JI,-X' ) JI,=I m=1 i=1 ~ ~ ~m
(3.6)
V#m + (~a.)
2
+ 4 ~ ~ a.a.I(x.n-x.
i"Fj
~
~J
~~
~m
)I(X.Q-X. J.
Jm
)]
which can be written as P ~
4
i=1
2 a. ~
n
n
~
~
JI,=I
m=1
I(X.JI,-X, ~
~m
)+2~~C
..
n
n
~
~
i"Fj ~J JI,=I m=1
n
2 P + 0': a.) n(n-I)-4(E a.) ~ a. ~ ~ i=1 ~
1/J •. (JI"m)+ ~J
n
~ JI,=I
~ I(X.JI,-x, ), m=1 ~ ~m
JI,"Fm
after using the relation
I/J .. (JI"m)
I(x.n-x. ~~
~J
~m
)I(X·n-X. J~
+ I(x.
-x.n)I(X.
-x. n ).
since
I(x.n-x.
)+I(x.
~m
Finally,
Jm
~~
~ ~
~m
Jm
)+
J~
~m
-x. n ) ~ ~
2
2n (n-I )~ a. +
p(p-I)n (n-I) T4
~
(3.7) p
+ 2
Comparing
p
n
2
n
C .. ~ ~ I/J .. (JI"m)-3n (n-I)(~a.) • ~ i=1 j=1 ~J JI.=I m=1 ~J Uj JI."#m ~
~
(3.3) and
linearly related to
(3.7),
it is clear that
T4
Hence the two tests are
T3 •
-
106 -
is
equivalent provided
C jk
=
aja k
S. j,k
(I
s.
p).
REMARK 3.1. Clearly all the statistics used are consistent estimators of their expectations. Hence if for a fixed alternative hypothesis ¢O
then the test based on (i = I ,2,3).
Thus
=
!P jk
!.
Hence
TI
~
is
HI'
EH (T,)-E ,
I
cons~stent
~
(T,)¢ 1!0
aga~nst
~
is consistent against any
TI
alternative for which
T,
~ ~ C'kP 'k¢O, j¢k J J
where
CJ'kPJ'k
is consistent against all altern-
atives for which at least one bivariate correlation is nonzero, provided the sign of this correlation is known. Similarly
T3
is consistent against all alternatives
for which
Tjk¢O
for some
j,k. Any reasonable test
should be consistent for a broad class of alternatives and preferably be optimal for certain specific alternatives and preferably optimal for certain specific alternatives. We have just seen that the tests proposed \ satisfy the first requirement. Towards the second ~equirement,
we shall prove in Section 5 among other
things that under certain assumptions, a test based on p p
~ ~ C'kS'k'
the statistic
j~k
J
J
where
is the Spearaman
S'k J
rank correlation coefficient between
j-th and
k-th
variates, in the sample, is locally most powerful against logistic alternatives. Notice that ~
S'k J
and
~
T'k J
are
asymptotically equivalent (see HAJEK and ZIDAK [14J, p. 61), which implies that in large samples a
T3
test based on
is locally most powerful against logistic altern-
atives.
4. ASYMPTOTIC RELATIVE EFFICIENCY Consider the sequence of Pitman alternatives given by (4. I)
x -
107 -
(L'>, •••
,L'»',
a
x*
where the components of and independent of
z
are mutually independent
which has a continuous distribu-
tions having zero mean and a finite third absolute moment.
is a non negative mixing constant and
L'>
may take values +1
or
We shall compute
-I.
Pitman efficiency of the test based on that based on
T j • Let
a in ,
bi~
aiO'
~
in particular,
are bounded constants which,
(i=I, ••• ,p)
a.
the
E .. ~J
relative to
T.
~
respectively be
HL'>' effective mean under
the effective mean under
and effective standard deviation under
HO
of
HO
T .• ~
Towards this, note that
E (x .x .1 H ~
x.
where
]
1/2 2 2 a.a.1:I (J In
)
x.]
and
~
n
is the variance of
~
]
are two components of Z.
aJ/.n
In
and
(J 2
It follows that p p
L'>2(J2
(4.2)
x
~ ~
p (p-I )
Uj
For the non null means of
2 2
a .a .' ~
]
and
T2
T3
we have the
following lemma.
LEMMA 4. I. If
f.
~
sities of the components of ponding distribution
x'"
functions)
(and
f'.'
ii)
tegrable with respect to 2 as f i (x) .... 0 Ix! .... d,
then under
exists,
F.
~
the corres-
satisfy
i)
~
the marginal den-
(i=I, . . . ,p)
is continuous and uniformly in-
HL'>'
-
108 -
Fi ,
(i=I, . . . ,p),
(I
(2)
RSa
T
In
:::;; p)
p (p-I )
2 2 2
S=~~
'f: k
2
2 a S'
--+ 2P- 1
where
:::;; j
2
-1/2) + 0 (n
2
a.a.6. Jf.dxJf.dx ~ ] ~ ]
i'f:j
PROOF. We shall prove special case and
(3)
since (I) follows as a
(2) becomes a simple corollary of
(3).
Consider and
= 2P(X*i ,Q.
\
=2 J z
m
where Z.
G
J z,Q.
are concordant)
X
-In
I/4, <_ x*i m + a.A(Z ~ L\ m -Zn)n!C p
n
i=1
[JF.(x+a.t..(z ~
~
m
-z,Q.)n
-1/4
I-I -
~
,
•••
,
p)
)dF.]dG(z,Q.)dG(z) ~ m
denotes the unknown distribution function of
Now using a Taylor expansion, n
2 J
c
Z
2 J
zm
m
J z,Q.
J z,Q.
p
n [ J {F. (x)+a .6.(z
i=1
~
~
-z,Q.)n m
-1/4
fi (x) +
nP [ I /2+a ~.6(zm.-z,Q.)!f~2.dx+o(n -1/2 )] dG(zm)dG(z,Q.)
i=1
-
109 -
since
ff~(x)dF. ~
2
ff.df. ~
~
O.
fdF.
2
~
~
Finally, we get
nc
0
2
p::J + 2 P - 3 n l / 2 2
222
LL a.a./), f f . f f . ~ ] ~ ] Hj
+ o(n- I / 2 )
after noting that
20
2
•
This completes the proof. Applying
(4.1) we get the following expressions for
the relative efficiencies when
=
f.(x) ~
E 23
o.
for
~
i=I, •.. ,p.
(4.3)
X-ll. f(----~)
(LLa.a.) ~ ] Hj
3 P- 2 2 P (2 P +I)-2·3 P
2
2 2
LL Hj
a.a. ~
]
and (4.4)
E31
1 44 (f f2 d x) 2 .
REMARK 4. I. When ai=l, E23 equals unity for and 3. This is expected since for p=2, T2 and identical and for
p=3
Also,
E23
explained by the fact
T2
increases with
are
two vectors can still only be
concordant or discordant so that information.
p=2
... 0
as
T2
gains all the which may be
P
that loss of information in using
p.
5. A LOCALLY HOST POWERFUL TEST FOR INDEPENDENCE IN MULTIVARIATE CASE 5. I. DERIVATION OF THE TEST.
HAJEK [13J has derived
a locally most powerful test of independence in the bivariate case which,
in particular,
includes
Bhuchongkul's normal scores test [4J, Elandt's [8J -
I 10 -
quadrant test and Spearsman's rank correlation test as special cases.
In the following, we derive the locally
most powerful test for the multi-variate case. Let X. = X':+a .6.Z ~
~
(i= I, ... ,p)
~
x~,
where x~
mutually independent and
...
,X;
are given bounded constants.
f. (x) , ~
and
G(x)
a.
~
In particular, they could
±I. Then, the joint density of
be
are
Z
has a known density
~
has an unknown distribution function
Z
and
is
p
n
(5. I )
f.(x.-a.6.z)dG(z). ~
i=1
~
~
(i=I, ... ,p). We
wish to test
>
=
HO:l:l
O. AIso, let
0
against the alternative
(x I k ' X 2 k ' •.. , X pk) ,
a random sample of size population. Let
R. = ~
n ~
denote the vector
~n
n
on the
i-th compon-
(i=I, ... ,p). Then, we have the following theorem.
Ez2 <
i=I, ... ,po
~
Then,
against
H6.
where
~
~~
i<j
K Rik
ex
is to reject
a.a.E ] ~
~
~
exist, be
~
f. (X ~
H'
when
)
~ (X ) ] Rjk,n ) f . (X ] Rjk,n
0
ik,n
)
Rik,n
is determined by
<
for
co
f
E
>
K
X ik
in the ordered
(k=I,2, ... ,n; i=I,2, ... ,p).
PROOF. The proof is not dissimilar to that of Hajek (1968). Consider
-
I I I
ex
the level of significance
ex,
denotes the rank of
(Xil"."X in )
f:'(x)
Jlf~(x) Idx
and
co
f ~ (X R
k=1
and
the locally most powerful rank test of
n
(5.2)
<
Jlf~(x)ldx ,
continuous,
and
>
(k = I , ... , n) den 0 t e
(R.I, ..• ,R. )'
THEOREM 5. I. Let
HO
6.
drawn from the multi-variate
of ranks of the sample of size
~nt
H6.:
-
lim
ll.
-2
[P (R .
·-1 , ... ,p.H I) - ( n.I)-P] A
r.,
-~
ll. .... O
~-
-~
•
L>
lim ll. .... O n
p
- n
n
n p f.ll. (x .k)} n II dX ik = k=1 i=1 ~ ~ k=1 i=1 n
1 i m ll.-2 ~ {
n
P
n
n
j=k+1
i=J
n
J
h ll. (x 1 ., ••• , x
R =r j=1 --p -p
k=1 ~I=EI
ll. .... O
X
k-I
J
]
.) X PJ
n f.ll.(x .. ) n ~ ~J k=1
Now supressing the subscript
of the
k
x ik
for the
time being, we have
P
J ... J [ n
=J
z
z zl
f.(x.-a.ll.z) ~
i=1
~
P
P
- n
i=1
P
n
f. (x .-a .ll.z .)]dG(z) ~
~
~
~
i=1
dG(z.) ~
i=1
P
J[ n
z
-
~
P
n
f.(x.-a.ll.z)~
~
~
f.(x.)]dG(z) ~
i=1
~
P
P
J ••• J [ n z 1
z
(x . -a .ll.z . ) -
f.
i=1
~
~
~
~
n
i=1
P
Now, expanding in powers of
ll.
P
H(ll.)
n
i=1
we have P
f.(x.-a.ll.z)~
-
~
~
-
n
i=1 112 -
f
(x.) i
~
P
f.
~
(x . )] ~
n dG (z . ) •
i=1
~
t,,2
+
t.H'(O)
2
~{H"(et.)-H"(O)}
+
H"(O)
T
2
(0
< e <
I),
where p
n
H' (0)
f! (x .)
p
i=1
1.
1.
~
f.(x.)
i=1
1.
1.
f. (x.) 1.
a.z 1.
1.
and
H" (0)
a.a .z 1.
2
]
+
(5.5)
+
f':
p
"\, -..2. a.z 2 2] ... i= I f i
1.
where the argument
\
for each of the
i= I
(x.-a.t.z.)1.
1.
1.
n
i= I
1.
Now integrating with respect
to
t. 2 T(Var z)[ ~ 1.. J.J. "'"
t.(x.)·= 1.
1
p
II f.(x i= I
1.
)]x 1.
f!(x.)C(x.) 1.
1.
]
]
f. (x . ) f . (x . ) 1.
1.
]
a
a 1.
+ j
]
where
51
x .-a .t"z 1.
1.
J{H"(M)-H"(O)}dG(z)
and -
f.(x.). 1.
1.
we obt
p
ht,(xl, ••• , x ) - II f p 1.= I
X
1.
p
n
II f
H'" (t.)
(5.6)
is
f.
Analogous expansion will hold for
(i=I, ••• ,p).
113 -
S
p
= J ••• J
2
Z
n
(H*"(9'6)-H*"(0)}
dG(z.). ~
i=1
P
62
Now dividing throughout by
and taking the limit
inside the integral signs we obtain
6
lim 6 ....0
-2
(n!)
-p
(P(R.=r., -~
-~
n ~
~ ~
k= I i<j
All that is left is the justification for taking the limit on
6
underneath the integral signs. Towards this,
consider
J
SI
R =r -p -p
n
p
n
n
k=1 i=1
dx.kl ~
S
n
J
IH"(9t.)-H"(O) I
R =r
k=1
-p -p
S
~
~
~
~
p
n
i=1
dx.kdG(x)S ~
J If~(x.)ldx. +
If~(x.)ldx.
J
Hj
n
~
-co
]
]
]
p
+ 2
~
i=1
J -CD
If':(x.) Idx. ~
~
~
<
co
where
x in H"(96) and H"(O) is replaced by x i ik (i=I, ••• ,p). An analogous bound holds for the integral
of
S2
after integrating with respect to the
first each on the range
(-co,co).
proof of Theorem 5.1.
-
114 -
xik's
This completes the
REMARK 5.1. Notice that for the bivariate case, HAJEK [I3J does not need the assumptions on REMARK 5.2. In particular, equal to
]
(or -1) if the components
+1
~
can be set to be
a.a. ~
f':(x).
X.
~
and
X. ]
are positively (or negatively) associated. Of course, we expect the
variates to have a certain internal
p
consistency, namely that with
and
X.
~
is positively associated
X. ]
then
are also positively
associated. Special Cases.
If the
fi
are normal, we get the
extension of the normal scores test to the case. If the
~
of Spearman's rho to the alent to
p-variate case and is equiv-
(see (2.6».
T3
p-variate
are logistic, we obtain the extension
f.
If the
are double expon-
fi
ential we obtain the extension of Elandt's [8J quadrant tes~
to the
p-variate case.
5.2. ASYMPTOTIC NORMALITY OF THE TEST UNDER
HO.
Consider n ~
T
k=I
~ ~
a.a.b (R.k,f.)b (R .k,f.) ~ ] n ~ ~ n ] ]
~
a.a.b (k,f.)b (R. k , ~ ] n ~ n ]
i<j
where b
n
(R.,f.) ~
Then n (5.7)
T
~
k=I
where
~
i<j
o
denotes the rank of
-
115 -
f;) -
when
has rank
k
Thus
(i=I, ••• ,n).
T
0
p
p
~ a.
~ a .kbn(R.k,f.)
j=2 ]
k=1
]
]]
j-I ~
where
a.b ~
i=1
n
That is
(k,f.). ~
P ~
T
a.T.
],n
j=2 ]
(5.8) n
o
~
T.
],n
a.kb (R ,f.). ] n jk ]
k= I
T. (j=2, ••• ,p) are mutually ], n "" "" By Hajek's Theorem (see HAJEK anc SIDAK
Thus, under
RO'
independent. [14] p.
168)
I(f.)
E[f~(X)/f.(X)]
]
=
is asymptotically normal provided 2 < (j=I, ••• ,p). Since T is
T.
],n
]
]
00
a linear combination of the constants,
T
and a. are bounded ],n ] is also asymptotically normal under RO' T.
Straight forward computations yield
(5.9)
E(T.
],n
=
no.b . ] n,]
a]. k
and
IRO) n
where
a. ]
n
~ n k=1
Var (T . I RO) ],n
b
(n-I)
.
n,]
-I
[
~
n
h=1
n
(5. I I)
2 -2 a·k-na.] k=l] ] ~
=
n
a.a .b ~
]
.bn,]. ,
n,~
and
-
n
n 2 ~ b (m,f.) m= I n ]
(5.10)
x
b (h,f.), and
116 -
]
-
-2
nb
.]x
n,]
5.3. NORMAL SCORES TEST FOR THE As pointed out earlier, when (i=I, •••
,p)
p-VARIATE CASE
f.
~
=
~«x-~.)/a.) ~
~
we obtain the normal scores test for the
p-variate case. Bhuchongkul [4J has extensively studied the asymptotic normality and the asymptotic efficiency of the normal scores test in the bivariate case. When f.=~(
x -
~
a
~
. ~)
p p ~
n
a.a. ~ n i<j ~ ] k=1
T
one can write
(i=I, ••• ,p)
~
E(z
Rjk,n
)
(5. 13) a .a .T . . , ]
~
~J
denote the standard normal (!l.=I, ••• ,n) whxe the Z !l.,n ord r statistics. BHUCHONGKUL [4J has shown that T . ., ~J
when suitably standardized,
is asymptotically normal
under all the hypotheses. Further, the asymptotic variance of
under
T.
~J
HO
n- I
is
(see [4J,
(4.1.5».
Hence p P
(5.14)
Var(TIH O )
~
n
2 2 a.a.
~
~
i<j
and the effective mean of
T
(5.15)
J
where and
~
~ij
J
X. ]
~
i<j
¢ and
-I
a.a. ~
]
J J
]
is given by
(i . (x) ) ~
J
(F . (Y) ]
llij
where
) d H .. (x, Y) ~J
, Hij is the joint distribution of Xi F. denotes the margianl distribution of ~
-
ll7 -
(i=I, ... ,p).
X.
~
Now let us consider Pitman alternatives
of the form: (5.16)
where
(6, . . . ,6),
X
z
6
and
X*
are mutually independent, and the
components of
X*
are mutually independent, having
F.
~
for the distribution of Xi' Also, let Ez = 0, Var Z = z has an unknown distribution function and = (J 2 < 00
G(z). Further assume that
~
continuously differentiable. on
z
and
has density
F.
f.
~
which is
Notice that the assumptions
are less restrictive here than in Lemma
f
4.1. Then, the effective mean of
is given by
T
~(6)
where (5.17)
~ ~
~(6)
a.a.~ .. (6). ~ ] ~J
i<j
Now consider
(5.18)
a . a . J J[J(F.(x»J(F.(y»h .. (x,y) ~
-
]
~
-00
J(F.(x»J(F.(y»f.(x)f.(y)]dxdy. ~
~
]
]
Now expanding the integrand about that
(5. 19)
~J
]
JJ(F.)dF. ~
~
=
0
for all
i,
6
=
0, and noting
we obtain
lim 6-0
(/~ ~ a~a~[ JJ(F.)f~dx][ JJ(F.)f~dx] i<j
~
]
~
-
~
118 -
]
]
X-jl.
f.(x) = f(----~)
Thus. when of
~
(5.20)
the efficacy
~
is given by
T
(i=I •.. .• p).
(J •
e(T)
Now one can evaluate the Pitman efficiencies of relative to the
T.
the efficacy of
TI
~
(5.21 )
J
which is equal to than unity when
when f
T
(i=1 •...• 4). In particular. since 'S 4",,,, 22.• we h ave 1 (J ~ ~ a.a i<j ~ ]
f
-I
is normal and is greater
is not normal. Thus
ymptotically optimal when the
=
F.
~
TI
is as-
are normal.
REFERENCES [I]\T.W. Anderson.
Introduction to multi-variate
analysis. John Wiley. New York.
[2]
C.B. Bell - P.J.
1958.
Smith. Some nonparameteric tests
for the multi-variate goodness of fit. multi-sample independence and symmetry problems. Multi-variate Analysis II. Ed.
New York. [3]
J.R.
P.R. Krishnaiah. Academic Press.
1969.
Blum - J. Kiefer - M. Rosenblatt. Distribution-
free tests of independence based on sample distribution function. Ann. Math. Statist .• 32(1961), 485-498. [4]
S. Bhuchongkul. A class of nonparametric tests for independence in bivariate populations. Ann. Math. Statist .• 35(1964).
138-149.
-
119 -
[5]
N. Blomquist, On a measure of dependence between two random variables, Ann. Math.
Statist.,
21(1950),
593-600. [6]
P.C. Consul, On the exact distribution of the likelihood ratio criteria for testing independence of sets of variates under null hypothesis, Ann. Math.
[7]
R.C. Elandt, A Acad.
[8]
38(1967),1160-1169.
Statist.,
Polon.
nonpa~ametric
Sci., Ser.
Sci.
test of tendency, Bull. Biol., 5(1957),
187-190.
R.C. Elandt, Exact and approximate power of the nonparametric test of tendency, Ann. Math.
Statist.,
33(1962), 471-481. [9]
D.A.S. Fraser, Nonparametric methods in statistics, Wiley, New York,
1957.
[10] D.V. Gokhale, On asymptotic efficiencies of a class of rank tests for independence of two variables, Ann.
[11]
w.
Inst.
Statist.
Math.,
20(1968), 255-261.
Hoeffding, A nonparametric test of independence,
Ann.
Math.
19(1948),546-557.
Statist.,
[12] W. Hoeffding, A class of statistics with asymptotically normal distributions, Ann. Math. Statist., 19(1948), 293-325. [13] J. Hajek, Locally most powerful rank test of independence, Studies in Math. Statist., Theory and Applications, Akad. Kiad6, Budapest, [14] J. Hajek - Z.
1968, 45-51.
Sidak, Theory of rank tests, Academic
Press, New York,
1967.
[15] D.N. Lawley, The estimation of factor loadings by the method of maximum likelihood, Proc. Roy. Soc. Edinburgh, 40(1940), 64-82.
-
120 -
[16J E.L. Lehmann, Some concepts of dependence, Ann. Math.
Statist.,
37(1966),
1137-1153.
[17J D.F. Morrison, Multi-variate statistical methods, McGraw Hill, New York,
1967.
[18J G.E. Noether, Elements of nonparametric statistics, Wiley, New York, [19J M.L. Puri - P.K.
1967. Sen - D.V. Gokhale, On a class of
rank order tests for independence in multi-variate distributions, Sankhya Ser. A., 32(1970), 271-198. Addendum. HO
The asymptotic normality of
under
follows from a result of Cramer (Mathematical Methods
in Statistics, Princetion Univ. Press,
1946, pp.
254-255).
Its asymptotic normality under all the hypothesis follows from a str~ht forward extension of a result of Cramer (1946, p. 366) pertaining to the asymptotic normality of functions of sample moments. Acknowledgements.
The authors thank the referee, and
Professors Bartfai and Vincze for a critical reading of the manuscript.
z.
Govindarajulu
Dept.
Stat., Univ. of Kentucky
857 Patterson Office Tower Lexington, KY 40506, USA A.P. Gore Maharasta Association for Cultivation of Science Poona, India
-
121 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
ON A GENERALIZATION OF STIRLING'S NUMBERS OF THE FIRST KIND B.
GYIRES
INTRODUCTION
(
In this paper quantities are defined which can be considered as a generalization of Stirling's numbers of the first kind
([IJ,
Chapter IV).
It will be shown that
these quantities and their generating functions certain partial difference equations Sec.
(Sec.
satisfy
1 and 2).
In
3 we give an explicite representation of these qua-
ntities and of their generating functions as well.
Then
the general results will be applied for the Stirling's numbers of the first kind.
It seems that in this way new
results are obtained for the Stirling's numbers of the first kind.
I.
THE DEFINITION OF THE
R-NUMBERS AND THEIR
FUNDAMENTAL PROPERTIES In this paragraph we investigate numbers, which allowed us a certain generalization of the Stirling's numbers of the first kind.
-
123
-
Let G(z)
z
~
k
k=1
be an analytic function, which is regular in the neighbourhood of
z=O, where the elements of the sequence
{ak}~=1
are complex numbers. DEFINITION I. The numbers n k - , -k\ (d G (z ) ) n. . d zn z= 0
(I)
are called to be
(k = I , 2 , ... , n;
n = I , 2 , ... )
R-numbers generated by the sequence
Let n G (z) n
ak
~
ak
00
H
n
~
(z)
tions
G(z)
function
H
z=o
and n
k
z
k
k=n+1
Obviously
z
k
k=1
, k
.
is a zero of order one of the func-
G (z) n
and the zero of order
n+1
of
(z), respectively. Therefore
o
(2 )
THEOREM I. The
(k=n+l,n+2, ... ).
R-numbers,
given by Definition
depend on the elements of the sequence
PROOF. Since -
124 -
n
{ak}k=1
I,
only.
k
n
~ (k.) (~ .... dz n
j=O ] and
z=o is a zero of order k-j G (z)H (z), we have
j Gn
k-j
(Z) Hn (») Z z=O
(k-j)n+k
of the function
j
n
n
o (j=O,I, ... ,k-I),
thus dn
k
(-n Gn[(Z»)z=O dz
which is the statement of Theo em I. THEOREM 2. Let
S i
<
k.
Then the partial diff-
erence equation
(n=2,3, ••• ) is satisfied by the {a k
);=
R-numbers generated by the sequence
I
PROOF. Applying the Leibnitz' formula on the right side of I
dn
i
kT(-n dz
k-i
G(z)G(z»)z=O
we obtain the relation
Using equality (2) we get
-
125 -
dV
R,
( - G(z»)z=O=O
dz v
(
(v=O,I, ... ,R,-I),
dn-v k-R, n-v G(z»)z=O
o
(n-v=O,I, •.. ,k-R,-I).
dz
carrying through a simple combinatoric modification we obtain the statement of the Theorem 2. We are aware of the circumstance that deserving case,
R
1
(v)
R,=I
is a
for which we have a
dV
1
= - ( - G(z»)
v! dzv
v v
z=o
even if this relation is a trivial consequence of the Definition I. Namely the following Corollary holds: COROLLARY I.
Rk(n)
(3)
=
The linear partial difference equation
n-I
a
~
k
v=k-I is satisfied by the
Rk _ 1 (v)
n-v n-v
(n=2,3, •.. )
R-numbers generated by the sequence
{ak};=I·
2. GENERATING FUNCTION OF THE For
n=I,2, ...
R-NUMBERS
we introduce the following genera-
ting functions: (4 )
-
126 -
THEOREM 3. (5 )
xn gn-I (x)
dn
eG(z)/X)
(x~O;
z=o
nr(dZ n
n=I,2, ... ).
PROOF. Using ( 1 ) we get n gn-I (x)
=
Taking the power
nT
! k=1
1
dn
k
rr(dZ n G(z»z=O x
n-k
out, then taking into consideration
xn
relation (2) and finally changing the order of the summation and derivation, we obtain that (x)
g
n-I
=
X:[d
n
n. dz n
(eG(Z)/X_ 1 )]
, z=O
which is already the expression (5). THEOREM 4. The partial difference equation n-I n-v-I ! an _v x gv-I (x), n v=O
(6)
(n=I,2, ... ; g_l(x)=I) is
satisfied by
the
generating functions
PROOF. On the basis of formula (x) gn-I
n = ~ n!
(d n-I dzn-I
(5)
(4) •
we obtain that
[ eG(z)/x G'(z»J x z=O·
Applying the Leibnitz' formula, we come to the expression n-I ( _d_ _ [ e G ( z ) / x G ' (z )] )
dz
n-I
-
z=o
127 -
v
n-I
= .:..,'"
(n-I)(_d_ eG(z)/X)
v=O
v
dz v
(d
n-v
x=O dz n - v
G(z)
)
z=O
.
Since v! v gv_l(x), x
(
dn - v
-n-:v dz
G(z))
z=O
(n-v-I)!a
=
n-v'
a simple calculation yields the relation (6).
In agree-
ment with Theorem 4 I.
Let
THEOREM 5. A solution of the partial difference equation (7)
(6)
is given by
gn-I (x)
1 -,D
n. n (al, •.. ,a n ; x)
-
128 -
(n=I,2, . . . ).
PROOF.
We begin with the system of equations
k-I
(8)
L
k-\)-I ak_vx
g\)_1 (x)-kgk_1 (x)
=
-akx
k-I
\)=1 (k=I, ..• ,n)
obtained from
with unknown functions
(6)
gk(x)
(k=O,I,
Substituting
. . . ,n-I).
(k=I, ••. ,n),
(9)
we get
gn_l(x)
a a a
n-I
n-2 '1-3
a a a
n-2 n-3 n-4
a a a
n-3 n-4
a2
a I
-a
a I
-(n-I)
-a
0
-a
-(n-2)
n-S
n n-I
n -2
X
as a
a2
a l
-3
0
0
-a 3
a I
-2
0
0
0
-a 2
-I
0
0
0
0
-a l
(8).
solution of the system of linear equations
Let perform the following modifications in the determinant on the right hand side of the Let us
intercharge the last column and
take out
the sign
multiplied by
Due to
-I.
this,
last equation.
the first
one and
the determinant is
Let us replace now the
a .'s using
(9). Multiplying now the k-th column
(k=I, .•. ,n)
(_I)n.
]
formula by
xk - I
get
for
k-th row,
and dividing simultaneously by x
n-k
the exponent expect of the
129
n(n-I)/2
,
in all elements of the
(n-k+2)-th one, -
x
-
in which the
we
exponent of xn- k
is equal to
x
from this row
gn-I (x)
a a a
nT a
n
a
n-I
a
n-2
n-k+l. Taking out the power
we obtain
(k=I, ••• ,n)
(_I)n(n-I)/2
a
n-I
a
n-2
a
n-3
a
n-2
a
n-3
a
n-4
X
n-3
a2
n-4
al
al -(n-I)x
-(n-2)x 0
n-S
X
a3
a2
al
-3x
0
0
a2
aI
-2x
0
0
0
al
-x
0
0
0
0
Finally replacing the inant by the
n-k+l-th one
k-th row of the last determ(k=I, ... ,n), we get the iden-
tity (7).
3. GENERALIZED STIRLING'S NUMBERS OF THE FIRST KIND Let
be a monoton increasing ordered
(il, ••• ,i k )
combination of order
of the elements
k
I , ... , n
without repetition. Let ( 10)
A
al
(al)A
a2
(a 2 ) ••• A
as
where A
aj
= (a., a.+I, ••• ,a.+a.-I),
(a.) J
J
a '+I-(a.+a.-I) J
J
J
J
~
J
2
-
J
(j=I, .•• ,s).
130 -
(a),
s
Obviously
al a
5
= iI' +0.
5
ik•
-1
DEFINITION 2.
If the representation (10) holds we
say that the combination the elements
(iI, ••• ,i k )
with lengths
k
of
without repetition and without
I, ... ,n
permutation is decomposed into blocks
... ,5)
of order
0.
1 , •••
A
aj
(a.) ]
(j= 1, •••
,0. 5 •
It is easy to see that this decomposition of is unique.
••• ,i k )
Let ( 11)
E
<· 1< ••• <.~ k-< n 1-~
i I i 2 ···i k a +I··· a +1 0. 1 as (k=I, ••• ,n; n=I,2, ••• ).
DEFINITION 3. The numbers (k= 1 , ••• , n;
n = 1 ,2, ••• )
are said to be generalized Stirling's numbers of the first kind generated by the sequence If
ak=I
(k=I,2, ••• )
then we obtain the well-known
Stirling's numbers of the first kind ([IJ, Chapter IV). THEOREM 6. D n
n n-I k (aI, ••• ,a ; x) .. a l + E ak(n-I)x, n
k=I
-
131
-
where D (al, ... ,a ; x) is the determinant introduced n n in Sec.2, and ak(n-I) (k=I, •.• ,n-I) are the numbers (11)
{ak}~=I.
generated by the sequence PROOF. Obviously
Dk + 1 (Y I ,··· ,Y k )
0
-Y I
0
o
0
0
0
-Y 2
o
0
0
0
0
o
ak + 1
0
0
o
( 12)
Denote by D
n
D.
~
I ...
.
~
the determinant obtained from
k
as follows.
(al, ... ,a ;x) n
D
n
(a l , . .. a ; n
< •.. <
ik are unchanged. The remaining rows are also unchanged except for the elements
x)
with indices
The rows of
i l
after the main diagonal, which will be zero. After a slight modification we get on the basis of
(12)
that
provided that the combination into blocks with lengths combination of
a l , . . . ,as' Summing up for all I, . . . ,n-I, without repetition and wtthout
permutation, we obtain the statement of Theorem
6.
We get the following Corollary from Theorem 6. COROLLARY 2.
The generating functions of the
generalized Stirling's numbers
Sk (n-I)
(k= I, ... ,n-I)
of the first kind generated by the sequence -
132 -
(ak}~=1
are the polynomials
D (al, ... ,a ; n n
-x)
(n=I,2, •.. ).
If we compare the results of Theorems 5 and 6, we get that the connection between
R-numbers generated by
and generalized Stirling's numbers
the sequence
of the first kind generated by the same sequence, is given by I n-k -(-I) S (n-I) n! n-k
( I 3)
(k=I, ... ,n-l; n=I,2, ... ), R
n
(n)
From this we obtain the following THEOREM 7. The quantities partial difference equation
(13) satisfy the linear
(3).
Let us denote (k=I, •••
,n;
n=I,2, •.. )
the Stirling's numbers of the first kind. Let in the case (k= I ,2 , ••• )
D (1, ••• ,1; x) n
=
D (x). n
Then we have the following Corollary from Theorem 6. COROLLARY 3. D
n
(-x)
n-I 1+ ~ s(n-I)x k k=1 k
-
133 -
(x-I) (x-2) ••• (x-n+I).
Taking into consideration formula
(13), we get the
following partial difference equation from Theorem 7. COROLLARY 4. _1_(_I)n-kS(n-l) n! n-k
I[ I
=k
n-k+1
I (k-I) !
+
( I 4) +
n-i 1:
(_I)V-k+1
v=k-2
VI
n-v
S (V-I)] k-I •
On the basis of Theorem 5 we obtain COROLLARY 5. n-I
D (-x) n
( 15)
If
GO(z)
(n-I)!
I v!
--(-x)
1: v=O
n-v-I
D
v
(-x).
is the well-known harmonic power series,
i . e. k z k
then we get the following two Corollaries using Definition I and Theorem 3: COROLLARY 6. (16)
(k=I, ••• ,n-l;
n=2,3, ••• ).
COROLLARY 7. n dn (-x)
(---e dz
-Go (z)/x
n
-
134 -
)
z=O'
(x;fO;n=I,2, ••• ).
The results
(14)-(17) concerning the Stirling's
numbers of the first kind and their generating functions seem to be new.
REFERENCE [IJ
Ch. Jordan, Calculus of finite differences, Rotting and Romwalter,
B.
Sopron,
1939.
Gyires
Mathematical Institute of the Kossuth Lajos University 4032 Debrecen, Pf.
10, Hungary
-
135 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION IffiTHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
CONSTANT REGRESSION OF QUADRATIC STATISTICS ON THE SAMPLE MEAN,II B. GYIRES
I.
INTRODUCTION ~I""';n
Let
be independent and identically
distributed nondegenerate random variables with finite second moment. Denote the characteristic function and the second characteri,tic (see, e.g. on distribution
=
log f(t)
o~
~I""'~n
[5J, 2.4.) of the comm-
f(t)
by
and
~(t)
respectively. Let n
n
~
~
j=1
k=1
a]'k~]'~k
be quadratic statistic with real coefficients and let A
=
~I+" '+~n'
Let n
c = E(O),
~
a .. ". 0,
j=1
n
n
~
~ a'k'
j=1
k=1
]]
b
]
-
137 -
Let the sequence
be defined by the
recurrence formula
c1
=
I,
(I)
k
C2k + 1
=
2k+1 V:IC2V-IC2(k-V)+I·
The author proved (see [IJ, p.52) the following and i f the random variable
Q
is an entire function,
f(t)
f(z)
(2)
=
A, then
has constant regression on and
(Izl
exp q>(z)
<
R)
where (3)
q>(z)
a
- "2
2 z -
~
k=2
C 2k - 1 kbk-I 2k a z 2k (a
with radius of convergence
0,
b
~
p/Iab, where
R
fixed number in the interval
>
[/2, 13]
the zeros of smallest modulus of
and
0)
p
is a
z = ±R
are
f(z).
Completing this theorem we prove the following main result of the present paper: THEOREM 2. Let
q>(z)
be the analytic function
defined by the power series (2)
(3).
Then the entire function
is a characteristic function i f and only i f
b
= o.
Starting from this Theorem we obtain the following result as a consequence of Theorem I.
-
138 -
~I'
THEOREM 3. Let
...
'~n
be independent and iden-
tically distributed random variables with finite second moment.
E(~j)
If
=
nB 1 ~ B2
and
0
>
0
then
Q
cannot
A.
have constant regression on
Theorem 3 was proved by the author in the special case
when
the
random
variables are infinitely divis-
ible
([IJ, Corollary 2). Theorem 3 excludes the possibil-
ity of characterization of distribution by the constant regression of quadratic statistics on the sample mean if nB 1
~
B2
>
O. This is also of interest in Theorem 2. As
far as we know similar theorems were not discussed in the literature. We deal with the proof of Theorem 2 in Sec. 2. The essence of the proof is as follows:
f(z)
function
First we expand the
into power series, then by the help of
this series show that all moments of order negative if and only if
b
4n
are non-
= o.
2. PROOF OF THEOREM 2 (a)
Let
f(z)
e
Ip(z)
.
Since (d
k
f(Ip») k z=O
(k=O, 1 ,2 , ••• ) ,
dip
we can apply the Faa di Bruno's expansion formula ([3J, 33), to get n
~ (2n)! k=O al+···+a =k n 1
~
-
139 -
(
d Zn
a
~
n
)-
(Zn) ! dz Zn z-O
Considering that
we get (-I)
n
a Zn
n
a
where ~
a l +· .. +an=k
a +ZaZ+ ... +na =n 1
n
c ••. x (
an (k=I, •.. ,n).
Zn-I) n
It is easy to see that the equalities cx1+ZcxZ+···+ncxn=n n=O. Thus
a l +·· .+an=O,
have common solution if and only if
bO(O)=1
and
bo(n)=O
if
n
~
I.
In the following we deal with the real roots of the polynomial n
(5)
gn-l (x)
=
~
bk(n)x
n-k
.
k=1
It is obvious that there are no positive roots.
If
n
is even then the number of the negative roots of (5) odd and there is at least one such root. On the other hand if ~
n
0, where
is even then necessarily MZn
denotes the
characteristic function
> 2nZn-th moment of the
M2 =(2n)!a n
f(t). We see at once that -
140 -
is
for (b)
large enough.
b
In this
section we construct representations of
the coefficients
(4) which can be handled easily when
Theorem 2 is proved.
We need the
Let
be a monotone increasing ordered
combination of order out
following notation:
of the elements
k
repetition and without permutation.
(6 )
A
a I
(al)A
a2
(a 2 ).
I, ... ,ll
with-
Let
•A
a
(a) ,
s
s
vlhe r e
A
aj
(a.,
(a.) ]
a.+I, . . . ,a.+a.-I), ] J ]
]
a. I-(a.+-a.-I) ] + ] ]
It
LS
th~t
obvious
k,
a l + ••• +a
If the
a
a
j
repr~sentation
comGination
(6)
s
+a
s
-I
holds, we say that the
is decomposed into blocks
(j= I It
(j=I, . . . ,s).
:::: 2
, ..
. , s)
with the lengths
is easy to see that the decomposition of combina-
Lions of order
k
of the elements
(I, ... ,ll)
withoUT
repetition and without permutation is unique. Returning to our duty we prove
the following state-
men t.
LEMMA
I.
ak(n) (7)
bn_k(n)
n!
(k=O, I , . . . , n- I
141
n=I,2, ... )
where 1•
(8)
(k=I ••••• n-I; n=I.2 •••• ).
and where the combination the elements
I ••••• n-I
(i I •...• i k ) of order k of without repetition and without
permutation is decomposed into blocks with lengths <X 2
a l •
····,a s · PROOF. Let n G
n
k
z .
~
(z)
k=I
According to k!
~
(
C2
1 a
n-) n z
1
a I !. .. a n ! (-1)
a)+ ..• +an=k .•• x
C
a l C <X2 3
(T)
a I +2a 2 +···+na
n
n
x ...
,
we get
i.e. the numbers
bk(n)
(k=I •••• ,n-I; n=I,2, ••• ) are
R-numbers generated by the sequence Definition 1
{e 2k - I };=I
([2J,
and Theorem 1). Thus the proof of Lemma 1
([2J, Theorem 5 and 6) is complete. (c) In this section we deal with inequalities. which will be used later. -
142 -
LEMMA 2. The sequence defined by the recursive formula
(1),
is strictly mon-
otone decreasing.
PROOF.
Since
C1
=
1
> 31 =
C 3 , we can suppose that
our Lemma is proved for the index
2k-1
(k
>
I).
If we
apply Theorem 368 of [4J in the formula (I), we obtain
which completes the proof. LEMMA 3. (k, R,=O , 1 ,2, ... ) •
PROOF. First we prove the left inequality. If
k=R,=O, then our Lemma is valid. Suppose that tru~
the Lemma is m+n
<
k+R,.
C2m + l , C2n + 1 with the recursive formula (I) we get
for the numbers
Apply~ng
n
C2 (k+R,+I)+1
2(k+R,+I)+1
{ ~ C2v-IC2(k+R,+I-v)~I+ v=1
(9)
k+R,+1 +
~
v=k+2
C2v-IC2(k+R,+I-v)+I+C2k+IC2R,+I}·
Since
(10)
k+R,-v+1 v
<
< k+R,+I,
k+R,+1
(v=I, ••• ,k),
(v-I
=
k+I, ••. ,k+R,)
holds, using the hypothesis, we have
-
143 -
thus
c 2 £+ I -zk+T
( I I )
k
L v=1
C
2v-1
C
2(k-v)+1
C 2k + 1 k+£+1 L C
C
2£+1 v=k+2 2(v-k-I)-1 2(£-(v-k-I))+1
( I2)
C 2k + 1 2£+1
£
L v=1
C
2v-1
C
2(t-v)+1
and finally on the basis of C
(9)
< C C 2(k+£+I)+1 2k+1 2£+1
and this is our statement. Now we turn our attention to the proof of the right inequality.
k=£=O
If
then our inequality is true.
that the inequality is true for numbers with
m+n
<
k+£.
Since formula
(10)
Suppose
C 2m + I '
C 2n + 1
is satisfied, making
use of our hypothesis we have
>2..c C 3 2k+1 2(v-k-I)-I·
C2v - 1 -
Using equalities
(II)
C2 (k+£+I)+1
~
and
(12) we obtain from
._=-I_.,---_ 2(k+£+I)+1
-
144 -
(9)
that
LEMMA 4. The inequality ( I 3)
a(n)
a I (n) (k=O, I, . . . ,n-2;
holds, the
provided
formula
a (n)
PROOF. hand side of
(8) =
that
the
numbers
ak(n)
n=3,4, ••• ) are defined by
and n-2
-9-
First we prove the inequality on the right (13).
According to
(8)
I
"3(1+2+ ... +(n-I))
n(n-I)
6
thus In-I "3 L j=1
L
I~il< •••
j i l •• .ikC2al+I·· .C2as + l •
a l +·· .+as=k
is the prod-
(a=I, . . . ,k)
uct of the elements of a combination of order the elements permutation.
I , ... , n-I
k+1
of
without repetition and without
Consequently the sum has
n-I) (k (n-I-k)
=
(n-I) k+1 (k+l)
products of the elements of combinations of order k+1 of elements
I , ... , n-I
permutation,
i.e.
appear
without repetition and without
all such products of combinations
(k+I)-times.
On the other hand since -
j
145 -
is different from
this new element
i l , ••• , i k
blocks with lengths
a
and
j
joins two neighbouring
8.
If on the left hand side
(or on the right hand side) this new element has not an immediate connection to a block of a=O
(or
then
( i l , · · · , i k ),
8=0). The length of the so arising new block
is equal to
a+8+1. Thus according to Lemma 3 we have
( 14)
i . e.
k+1
~ -3-
a1a k
L
< < . < -I 1<· ~ I ... ~k+l~
il···i
k+
IC
Za 1 + 1
••• C Z
as
+I~
a 1+ ... +a s =k+ 1 ~
(k=Z,3, ••• ,n-Z)
ak + 1
It is trivial that the inequality in the case
k=1
is satisfied. Now we turn to the proof of the inequality on the right hand side of (13). We start from the equality ( 15)
ak + 1
=
A+B,
where
A
(n-I)
.
I~ 1<"
L.
'<~k5n-Z
i I " .ikCZal+I·· 'C Zas + 1
al+···+as=k+1 B
If we wanted to modificate the expression that it contains the members of
-
146 -
ak ,
A
such
then we should apply
the procedure used in proving the left hand side of inequality (13), i.e. according to formula should multiply the members of
A
(14), we
by the factors
Thus conforming to Lemma 3
n-I
A ~ -3-
(16)
~
ISi I <.• •
i l •• .ik C2Cl+ I •• .C2Cl +1 s
Cl I + ••• +Cl s =k
We want now to modificate the sum
B
such that the
products of the elements of combinations of order elements
I, ... ,n-I
of
without repetition and without
permutation, containing the element among the members of
k
n-I, take place
To reach this we replace two
B.
factors of all members of
B
by
the product of these factors is
n-I. The minimum of 2. In this procedure
each block can be decomposed into two blocks twice, and two blocks can join into one block once. Finally if we form combinations of order
k-I
of elements
1, •.• ,n-2
without repetition and without permutation from all combinations of order
k+1
of the same elements, with-
out repetition and without permutation, then all combinations of order
k-I
will appear
we get 2 x
;Z (17)
-
147 -
(n;k)
times. Finally
Since
2 ;Z
_I_(k+ I) n- I 2
(n-k)
n-2
~ -9-
2
=
a (n)
n=3,4, ... )
(k=I, ... ,n-l;
and n-I 3a(n)
using (16),
>
(17) and
of the inequality
k=O
for
(d)
(n=3,4, ... ),
I
we obtain the right hand side
(IS)
(13)
taking into consideration,
that
this inequality obvious. In this section the main theorem of this paper,
Theorem 2 will be proved.
LEMMA 5, Consider the equation
n!
(5)
in the form
0,
gn-I (x)
where the coefficients are defined by (8).
• the
even.
Then
one -
are in the interval
real roots of the equation -
(I
a(n)'
-
>
0,
if
x
>
-
I a I (n)
gn-I (x)
<
0,
if
x
<
-
I a(n)
I)
a l (n)
least one negative root.
n
n
be
at least
and
,
It is obvious that equation
no positive roots, but since of
-
gn_l(x)
PROOF.
Let
is even,
g
l(x)=O has nthere is at
It is obvious that the moduli
the negative roots are positive roots of the equa-
148 -
tion
g
n-
n! ( I 8)
I (-x)=O. g
n-I
Since
n
is even
(-x)
therefore, by using Lemma 4, we obtain
( I 9)
n! where
gn-I (-x) < (l-ax)h n _ 2 (x),
a=a(n)
x
>
0,
and
One can see that
(19)
implies the statement of Lemma 5.
REMARK. Let us denote the roots of polynomial (18) Then
by
n-I
Using the inequality 2n - 1 (n-I) !
(.!.)
n-I
<
3
< I Z I··· zn_1 I <
C 2n _ 1
< (.!.) 2
3n - 1 (n-I) !
,
we get
,
or 2 n-I';(n_l) !
<
n-I
.; IZI·· .zn_1 1
Thus
-
149 -
<
3 n-I';(n
I) !
PROOF OF THEOREM 2. We have mentioned in (a) that if
is a characteristic function, and
f(t)
is even,
n
then the inequality
=
M2n =(2n)!a 2n
holds. Here
M2n
~) ~
(2n) !gn-I (-
is the
2n-th moment of
0 f(t).
According to Lemma 5
(
gn-I -
Since b
>
!!..) < 2 ~s
a(n)
0
0, n -
if
tion if
n, i.e. b
>
f(t)
2
I
a (n)
00, it cannot exist a number
such that inequality
all even
!!.. >
M2n
~
0
is satisfied for
cannot be a characteristic func-
O.
Thus the proof of Theorem 2 is completed.
REFERENCES [I]
B. Gyires, Constant regression of quadratic statistics on the sample mean, Analysis Mathematica, 3(1977), 51-53.
[2]
B.
Gyires, On generalization of Stirling's numbers
of the first kind, this volume, [3]
Ch. Jordan, Calculus of finite differences, Rotting and Romwalter, Sopron,
[']
1939.
G.H. Hardy, - J.E. Littlewood - G. P6lya, Inequalities, Cambridge Univ. Press, Cambridge,
[5]
E. Lukacs, Characteristic functions, Co., London,
1960. -
150 -
1952.
Hafner Publ.
B. Gyires Mathematical Institute of the Kossuth Lajos UI 4032 Debrecen, Pf.
10, Hungary
-
151
-
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
MARKOV-TYPE INEQUALITIES AND ENTIRE FUNCTIONS W.K.
HAYMAN -
I. VINCZE
INTRODUCTION
I.
In his paper [7J K.
SARKADI proved the following
rule of dualism for the Poisson distribution: P(K ~ k+1
peA < 9,IK=k) where
[A=9,)
is a random variable distributed according to
K
the Poisson law having expected value
A, while
distributed uniformly over the half line proof is based on the to A.
c~nditional
(0,00).
A
is
His
probability field due
RENYI [6J. As VINCZE remarked,
result of SARKADI,
i.e.
the real background of the the crucial point of his proof
is implied by the fact that the Poisson probabilities form probability distributions for both of the parameters: ~
(0
~
A
n=O -
153 -
< (0),
(n=0,1,2, ... ).
Then RtNYI and VINCZE formulated the following conjecture (see also [4] , problem 2.32) • Let 2
l+a l t+a 2 t + ••.
f(t)
(I)
(a i
~
0·, i=I,2, ..• )
be entire function for which (2)
(n=O, I ,2, •.. )
_ et •
f(t)
hold. Then
HALL and WILLIAMSON [2] have recently proved that
f(t)
if the entire function f ' (t)
1+ O( I)
f(t) as
t -
satisfies (I) and (2), then
logt '
co.
Their proof of this statement is based on r rather deep Tauberian remainder theorem due to G. FREUD
[I].
The present authors in their paper [5] sharpened this statement by elementary methods, proving that for arbitrary (3)
t
>
If'(t) f(t)
°
A > 56
and
-II <
A(I+t)-1/2
from which the relation e
follows.
t-2AIt
< f(t) <
e
t+2AIt
(t
>
0)
It was furthermore proved, that if
-
154 -
logf (t) then
f( t)
:::
e
t
t+ o( I )
(t -
00)
.
In the next section some arguments and results will be considered which occur in the mentioned article [5J but formulated in the language of probability theory. Our aim is to show how
elementa~y
tools of the theory of
entire functions can be applied in probability theory. 2. AN EXTENDED FORM OF THE INEQUALITY OF MARKOV ~(t)
Let half line
be a density function defined over the
(0,00)
for which we assume that it is contin-
uous and positive on the whole half-axis: (4 )
>
~(t)
~(t)
furthermore
<
(0
0
t
<
00)
has moments of all positive orders:
00
(5 )
o < J o
tn~(t)dt
<
(n= I ,2, .•• ) •
00
LEMMA I. I f
and
00
(6 )
J o
a
n
tn~(t)dt
(n=I,2, ••• )
then 00 f( t)
is an entire function.
PROOF. MUltiplying (6) by respect to
n, we obtain
-
155 -
An
and summing with
00
f f(H)cp(t)dt
(7)
<
(0 S A
I).
o
Hence for all positive
t
and
~ X+~
o <
~
f(AX)
min
f f(At)cp(t)dt < I-A
cp(t) S
xStSx+~
Thus
f(x)
x
a
LEMMA 2.
<
a n-I n+1
a
2 n
(n=I,2, ... ).
x
PROOF. We have for all real 00
o < f
t
11-
o
Thus
-2
an
<
I
2
x
(t-x) cp(t)dt
and 2
n 2x + a
-I
, which proves our statement.
CPO(t)=cp(t),
cp
n
(t)=a tncp(t)
-1,2, ••. ). For the expected value of co
00
= f tcp (t)dt = a f t non n 0 )
and denote by
n
cp
a random variable having the density
E(~
~
n
(a n- I a n+ I)
We write ~n
x.
converges for all positive
n+1
~n
n
(t)
(n =
we have
cp(t)dt
Now we formulate the following extended form of the inequality of Markov: THEOREM I.
Let
k
>
I,
then
(8)
(n S N)
(9)
(n
-
156 -
>
N)
aN_I
increases aN is an entire func-
PROOF. According to Lemma 2 rN strictly for tion,
...
rN
N
~
I,
as
00
rN :s; r
f(t)
and since
N ...
Further if
00.
:s; r N + I
f(t)
then the maximum term of the power series Consequently if
k
>
I, kr N :s; kr :s; t
and
is we
n:S; N
have
< _ k-(N-n)
aN
tN
.
Thus 00 00
f
(10)
kr N
This proves
(8)
with
Next we take k
n
N-I
>
instead of
N.
and deduce that for
N
t
:s; k
> N aNt kn-
N
Thus k
-I
r N+ I
f o
This proves
k
a tn
(9)
=
r N+ I
f
o
and completes the proof of Theorem I.
Let us introduce quantities d
-I
inf f(x)
157 -
d
and
D
as
-I
r,
D
sup
f(x)~(x)
O<x
for which we suppose that
We turn to the
00.
proof of the following THEOREM 2. We have for
N
~
0
~(N+3-2IN+2) < E(SN) < ~(N+2+2IN+I).
(II)
f(t), which sat-
REMARK I. As for entire functions ~(t)=[f(t)J
isfy (2),
-I
f(t)~(t)=1
and
hold, we have
The corresponding relation played an essential
d=D=I.
role in proving (3). REMARK 2. The assumption ~(O)=O,
when with for
a.
N
> 0 > NO'
d
>
0
does not hold e.g.
but using the definition
d =
inf
a.
a.<x
f(x)~(x)
we can obtain a similar statement at least the latter depending on
PROOF. According to
(8)
a.
as well.
in Theorem I for
k
>
n S N
f kr N+ I Writing
we deduce
Also
kr N + I
f o
kr N + I fN(t)~(t)dt
<
-
f o 158 -
f(t)~(t)dt
< Dkr N+ I
I and
Using (6) we have
we deduce N+I < Dr 1+ 2iDr N+ 1 N+I n+ from which the left hand side of (I I) follows. k=I+I/iDr
Choosing
To prove the right hand inequality in Theorem I, we dr N+ 1
assume that
>
I, since otherwise the inequality
is trivial. We can choose k
so deduce for
>
n k
-I
k
r N+ 1
J
I;
N r N+ 1
J o
a tncp(t)dt n
o
n=N+1
-I
(f(t)-fN(t»cp(t)dt
<
00
<
I;
k-I
n=N+1
'
and k
N+I
-I
>
r N+ 1
J
o
k
fN(t)cp(t)dt
This proves (II).
-
159 -
~
-I
r N+ 1
J
o
f(t)cp(t)dt
- _1_> k-I
3.
f(t)
SOME REMARKS ON THE CASE WHEN
DIFFERS
FROM AN ENTIRE FUNCTION In the above investigations we started with the ~(t)
assumption that the density function of all positive orders. N
J
t
has moments
Suppose now that for N +1
00
O~(t)dt
<
o
00,
while J t 0
NO
>
~(t)dt
00.
o
> NO' and the power series (t) . But even in this case f NO n Lemma 2 and Theorem I will hold, at least for indices a =0 for n ' reduces to a polynomial
n
In this case
and
N
less than
NO-I.
On the other hand if we assume the existence of all moments but drop assumption (4) the power series may converge in a finite interval
0
~
t
~
only. This is
R
the case e.g. when
~ (t)
--II' 0,
o
~
t
~
otherwise
for which ~ (n+l)t n n=O
f(t) i.e. .... R,
R=I. as
In such cases we can claim that
n ....
Finally we remark that the power series belonging to
~(t)
f(t)
plays a role in the moment constant
method of summation of divergent series p.
E(;n)=r n + 1
00.
81).
-
160 -
(see e.g.
[3J,
4. AN INEQUALITY FOR THE QUANTITIES
r
n
According to Lemma 2 for the quantities the inequalities
rn < r n + 1
(n=I,2, ..• )
r
hold.
n
an Imposing
~(t)
a further condition on the density function
we
are going to prove the following THEOREM 3. Using the notations and assumptions ~(t)
concerning the density function quantities as in nonincreasing with ~
t
<
(0)
and the related
§.2, assuming further that ~'(t)
first derivative
~(t)
~
0
u
=
(0
is ~
the inequality r
(12)
~
n
1_
r n+ 1
_....;1'-----,;-
(n=I,2, ... )
(n+ I) 2
holds.
PROOF.
Introducing the new variable
>..t
under
the integral sign in (7) we obtain (0<>"<1). Differentiation in both sides yields 00
-J
o
f(u)~'(~)~
=
= ___1__
du
>.. >..2
(1_>..)2
Returning to the original variable (13)
-J o
f(At)t~'(t)dt
Using the notation
(0<>..<1). u = t/>..
we have
(O~>"
~*(t)
-t~'(t)
(13) can be written
in the form {O -
161
-
~
>.. < I).
This means that the relations 00
f 0
a
n n+1
t n cp1'(t)dt
(n=0.1.2 •••• )
hold. from which we have 00
f
f1' (At) cp1' (t)dt
I-A
0
with
f'~ (t)
is the entire function associated with
the sense of (7) having coefficients
a'"
a'"
According to Lemma 2 for .~
r"
n
<
.', r"
n+1
Let
n
=~ a"
r'"n
cp*(t)
in
an
= -n+1
the relation
n
is valid. which is equivalent to (12). f(t)
be an entire function satisfying (I)
and (2) • In this case
cp(t)
Theorem 3 is valid. Le.
I
=
cp'(t)
f(t)
<
O. hence
(12) holds. For proving our :5
conjecture would need
I
__ 1_
n+1
REFERENCES [IJ
G. FREUD, Restglied eines Tauberschen Satzes I, Acta Math.
[2J
Hungar.,
2(1951), 299-308.
J.
London Math.
Soc.,
12(1976), 133-136.
G.H. HARDY, Divergent series, Clarendom Press, Oxford,
[4J
Sci.
R.R. HALL-J.H. WILLIAMSON, On a certain functional equation,
[3J
Acad.
1949.
W.K. HAYMAN, Research problems in function theory, Athlone, London,
1967.
-
162 -
[5J
W.K. HAYMAN-I. VINCZE, A problem on entire functions (in print in the volume to honour of I.N. Vekua on his 70th birthday).
[6J
A. RENYI, On a new axiomatic theory of probability, Acta Math. Acad. Sci.
[7J
Hungar.,
6(1965), 285-335.
K. SARKADI, A rule of dualism in mathematical statistics, Acta Math. Acad. Sci. Hungar., 9(1960), 83-92.
W.K. Hayman Imperial College London SW7, England I. Vincze Mathematical Institute of the Hungarian Academy of Sci. 1053 Budapest, Re!ltanoda u.
-
13-15, Hungary.
163 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21.
ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
INFINITE DIVISIBILITY OF DISCRETE DISTRIBUTIONS III S . K.
I.
KAT T I ,',
INTRODUCTION
A random variable is said to be infinitely divisible
~(t)
if its characteristic function
is such that
~I/n(t)
is a characteristic function of a bonafide probability distribution for every
n.
In non-technical terms what
it means is that there exist independent and identically distributed random variables n
that the distribution of
~ x
x
.
nl.
.
(i=1,2"
.. ,n)
such
is the same as the given
i= 1 nl.
distribution.
In agriculture, we usually start with a
large plot of land and then we subdivide it into aud assume that the yields in the
n-parts are indepen-
dent and since they are usually of equal size, identical distribution.
Thus,
n-parts
they have
if the distribution of
yield over the entire plot is such as to permit this subdivision,
then it needs to be divisible.
The question
This research was supported in full by a summer research grant from the Graduate School, University of Missouri, Columbia. -
165 -
of infiniteness in divisibility is really a theoretical idealization of the fact that the plot is subdivided quite extensively.
Same is true in biology where a blood
sample taken from a patient is subdivided into a number of parts to test for different diseases.
In testing for
reliability of equipment over a period of time, we take an interval and subdivide the interval into many parts and act as though the distribution of failure in each part is independently and identically distributed. This again leads to infinite divisibility. Hence testing for infinite divisibility is an important part of statistical inference. If one considers the full range of distributions, discrete, continuous and otherwise, then one does not come across an explicit test. Most such tests are of the form indicated by the Levy-Khinchine canonical form: The function
~(t)
is an infinitely divisible
characteristic function if and only if it can be written in the canonical form: . I 2 (e~tx_I_ ~tx )~ dg(x), 2 2 I+x x
+00.
ita+ f
log H t)
-00
where
i
=
~
and
a
is a real number and
non-decreasing bounded function with
g(x)
is a
g(-oo)=O. Note that
the problem of finding or deciding on the existence of the
g-function is left open. We found that by restricting ourselves to non-neg-
ative integer valued random variables, we get more specific results: Suppose that 0,1, •••
with
Po~O
PO,P I , ...
are probabilities of
PI~O.
and
Denote by
Pj
the ratio
Pj/Po. Then a necessary and sufficient condition for the inf~nite
divisibility is that for each
-
166 -
j.
=
g.
( 1)
]
jP~ ]
p*
j-I
q
1
-
Note that given a distribution function, one can numerically compute a number of
to see if they are pos-
qj
itive and if they are, then one can use this information along with his algebraic dexterity to generate an inductive proof of infinite divisibility. The .aim of this paper is to derive additional results. The above result and a few consequences are given in KATTI
[I]
and KATTI and WARDE [2].
2. DERIVATION OF FORMULA (I) USING THE LEVY-KHINCHINE FORM Since the random variable takes on the integer values
0,1, ... ,
g(x)
needs to be a step function also.
Hence, the canonical form reduces to 10g(
(2)
~ j=O
~ j=O
l+j2
Set t,
p.e i t j ) = ita+ ]
-.-2- gj" Differentiate ] it
then set
z=e
~ ijp .zj
j= 1
]
=
~ j=1
(2) once with respect to
~ P.zj){ia+ ~ (ijz j - -ij - 2 ) q.}. j=O ]
j=1
I+j
]
o
z , we get
j 1 +]·2 q].•
On equating the coefficients of other powers of follows.
j
co
(
On equating the coefficient of a
I+j
and rearrange terms to get
co
(3)
(eitj_l_ i t j )1+j2 2 2 gj"
Since the original
ing, it follows that the
g
z,
q's must be non-negative.
-
167 -
(I)
was monotone non-decreas-
3. SOME SUFFICIENT BUT NOT NECESSARY CONDITIONS Since any set of
q.'s being non-negative can act as ]
a set of necessary conditions, our attention is focused on getting sufficient conditions. The necessary and sufficient conditions are sufficiently complicated so that it is hoped that the availability of sufficient conditions will help users to use them to decide on the infinite divisibility of their specific distributions. A distribution satisfying the sufficient condition immediately implies that the distribution is infinitely divisible. Of course,
it not satisfying the sufficient condi-
tion does not imply that it is not so infinitely divisible. On rewriting equation (1) once with the original and then with
j+I
in the place of
j
j, we get:
(4 )
and
On mUltiplying (4) by
p'" Ip;' j+I j
and subtracting from (5),
we get p;' p,-, j - I j+ 1
p,-, j
(6)
Here
p~
o
is arbitrarily defined as unity.
p,-,
is monotone increasing, then -
p*.
.
J-~
168 -
----1!-!p" j
If
-p'~j-i+I
p
]
.Ip.] - 1
is
positive for each
i. Hence
being monotone I} J Jnon-decreasing is a sufficient condition for qj to be {P./P.
non-negative. This result was proved a different way in KATT! and WARDE [2J. To get a more stringent condition, we multiply (4) by
p'"
j+1
j+ I
and subtract
-p,', j
j
,',
p'"
j+1
p'"
j+1 j+1 p,': -Pj1']+ ••• +qi [," Pj - i ]
ql [ p. 1 - . JJ
+ ••• +qj
to get
(5)
J
j+1
p,':
,', -Pj - i
J
p~':
r]j+1
j+1 "'] p,': -PI· J
Thus another sufficient condition is: p,"
p'" j-i
j+ I
j+1
•
p,', j
j
p'~
~
j-i+1
(for all
i
<
j).
p'"
Denote
j+1 ---pi<
r.< Then, this condition can be written as J
j
j~1 J
r i+ I
~
U<j).
rJ'_;+1 ~
This condition is more stringent than the previous because the previous condition implies this while this does not necessarily imply the previous condition. To get another sufficient condition, we rewrite formula
(6) with
q
j+1
for
j
to get
-p'" +q j+2- j+2 1
-
169 -
]
+1
(7) p{'p'~
+q],+ I [
~*j+2 -p~]
•
j+1
On multiplying (6) by
p*
j+2
/p{,
and subtracting from
j+1
(7), we get p* p* j - I j+2] +
p':]
p'~j+2
-pr-
] +
j
Let
p,/p,
]]-
I
=
r,. Then a sufficient condition is that ]
each term on the right hand side be positive which leads to the set 6f sufficient conditions: r , 2 r ] '+ I ~ 0 2r ] '+2 r ]-~ , I r, , ,-r, ]-~+ ] - I- ]+
(i<j).
4. TESTING FOR INFINITE DIVISIBILITY From equation (I), we get condition ~
q2
~ 0
q2
= 2P~_p~2.
reduces to the condition
Hence the
2P2PO-P~ ~
O. Let
xI, ••• ,x n be a random sample from the integer valued distribution with f, standing for the frequency ~
~
fi
i. Let p, = -- • Consider the statistic ~ n A _ ..... 2 s = 2P 2 P O-P I • This statistic is an asymptotically consis2 tent estimate of 2P2PO-PI. The asymptotic variance is
of count
-
170 -
derived as below:
hence, asymptotic variance
V(s) A
A
+ 2P OP 2 cov(P O ,P 2 )-2P OP 1 cov(P I 'P 2 ) -
- 2P 1 P 2 cov(PO,P I )} P.(I-P.)
where
V(P.) =
~
~
n
~
P.P.
and
i*j, A consistent estimate
stituting
A
P.
~
for
cov(P.,P.) ~
v(s)
]
n
s
a-level test, and declare that
It>(s) the necessary condition was not satisfied if is the upper
where
for
is obtained by sub-
Pi' Thus to make an
we propose that the user compute
- ~
z
>
zo
a-percentile of the standard
normal distribution. REFERENCES [I]
S.K. Katti, Infinite divisibility of discrete distributions, Ann. Math. Statist., 38(1967),
1306-
-1308. [2]
S.K. Katti - W. Warde, Infinite divisibility of discrete distributions II, Ann. Matho Statist., 42(1971),
1088-1090.
S.K. Katti University of Missouri 222 Math. Sci. Building, Columbia MO 652 I I , USA -
171 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
AN EMPIRICAL GRAPH FOR CHOOSING A POPULATION DISTRIBUTION USING A CHARACTERIZATION PROPERTY S.K. KATTI - J.
1.
STITH
INTRODUCTION
Given a sample from any of a variety of theoretical distributions, the problem of choosing one for the analysis of the given data has long been a mojor concern to both the
theoreti~al
Historically,
thi~
and the applied statistician.
problem can be traced back to the
late nineteenth century.
In 1889, Galton used the normal
distribution to analyize the data presented in his Natural Inheritance.
However, in the next decade, Karl
Pearson found the normal inadequate in explaining his observations taken on animals and plants. This led Pearson to a study of distributional properties based on popu1ation moments. A basic approach for additional studies has been to group distributions of interest into a class. Once this class is formed, a method of distinguishing the members is sought. This is done with an eye towards using an equivalent method to classify samples taken from the -
173 -
class members into their proper distributional forms. A few
ex~mples
may illustrate this procedure. Egon
PEARSON defined a class of continuous distributions based on differential equations. Notable members are the normal, the beta, and the Student's
distributions. Discrim-
t
ination between members of this class has been based upon
al
and
a2 ,
functions of the third and fourth
population moments. Of the many papers written on this method, an excellant summary can be found in a paper by PEARSON and PLEASE [5J. Graphs distinguishing members of the Pearson system and several other interesting distributions are included. However, the authors note that their preliminary results using sample estimates of
al
and
a2 ,
bl
were not promising.
BOWMAN and SHENTON [IJ, on the other hand, considered bl and b 2 that modified previous work by D'AGOSTINO and PEARSON. A Monte Carlo study was functions of
performed and test contours were given to isolate the more appropriate distributions. A discrete analogue to the above work was carried out by J.K. ORD.
In his first paper [3J, he defines a
discrete class using a difference equation, then graphically distinguishes its members using
al
a2 .
and
In
the next paper, Ord [4J finds a new means of distinguishing his distributions and uses a sample equivalent to determine which class member best fits certain data. In Figure I, he graphs
r
against
f, r
random variable under consideration and relationship between
r
and
f
r
where f
r
R
is the
=P(R=r). The
forms the basis for
his work. This paper continues in this general mode. A class of longtailed,
symmetric distributions is defined by
ROGERS and TUKEY [6J. Noteworthy members are the normal, the logarithmic, and the Cauchy distributions. The -
174 -
u
f r
r---~~--~~--~~~------------~~~-----Poisson
Beta-binomial
Ilypergeometric ~------------------------~-r----_~~
Beta-Pascal Negative binomial
f
u
Poisson
r
Beta-binom~al
r----_~~
Figure
I.
Diagram showing the general shape of the curves
u =rf If r
r
-
r-
I
175 -
for various distributions
authors distinguish its members by use of their true density functions.
We will distinguish class members
given only sample informations. is
Since sample information
incomplete, this involves an additional level of com-
plexity. The method of classification used here was introduced by McDONALD and KATTI [2J.
Basically, they
consider the relationship between the sample mean and variance as a method of describing distributions. Independence of these variables characterizes the normal, while an empirical investigation graphically exhibits striking differences for two other distributions,
the
chi-square with two degrees of freedom and the Cauchy. These three distributions will be investigated so that a sample available in a particular problem might be classified as normal-tailed,
skewed or long-tailed. This
classification may suggest additional transformations to achieve normality or homoscedasticity. Several questions arise here. Do other long-tailed, symmetric distributions have graphs similar to the Cauchy? Also, can distributions with long tails, but not quite as long as the Cauchy's, be distinguished from the normal? Finally, can a testing procedure be formed using the graphs as a basis? Answering such questions is the focus of this paper. To resolve the first two,
an empirical examination of
Rogers and Tukey's class should prove satisfactory. As for the last, attention is restricted to the normal, chi-square and Cauchy distributions.
Here, if the test
statistic indicateE independence of sample mean and variance, the data is classified as normal-tailed. Otherwise, that is, if the test statistic fails to support such independence, the population is classified either skewed or long-tailed according to test results. -
176 -
2. THE TUKEY-ROGERS SYSTEM OF DISTRIBUTIONS ROGERS and TUKEY [6J describe a class of long-tailed,
symmetric distributions. This class consisted of
ratios, x=wjV
where
distribution,
V
chosen for
generate the various distributions of
V
W
follows the standard normal
is independent of
wand the forms
x.
Additional distributions are obtained through mixing as explained below. A few members have special names, while others are referred to by their symbols. The members of the class of distributions of
X
under consideration are given in Table I. TABLE I. A list of Tukey-Rogers system of distributions Symbol C
Name
Explanation
Cauchy
V
is standard normal.
G
Standard Normal
V=I
S
Slash
V
with probability one.
S/4
S/4
A probability mixture of Z5%
is uniform on
(0, I).
Sand 75% G, corrEsponding to
V=I
and
V
with probability 0.75 uniform on (0,1)
with probability Q
V
Q
0.25.
is distributed triang-
ularly on
(0, I)
with van-
ishing density at
QS
QS
O.
Q
A 50%-50% mixture of S, corresponding to
V
and trap-
ezoidally (altitude 0.5 at 0, 1.5 at (0, I ).
-
177 -
I)
distributed on
Name
Symbol
Explanation
35/4
3S/4
A probability mixture where
V=l
with probability 0.75 and uniform on
V
(0,1/3)
with
probability 0.25.
V=I
lOG/4
lOG/4
with probability 0.75 and
V=IO
0.25.
This distribution is approx-
Logistic
L
with probability
imated by a member. It is constructed by taking
Gucumatz
GUCU
an appropriate fraction (about 69%) of the standard normal between
and
-I
+1
and another appropriate fraction (about
105%) of that part
of the Cauchy outside these limits. In view of the complexity of the construction of this family,
it was necessary to verify that the
distributions we had obtained through our interpretations were the same as theirs.
Although this was virtually
unnecessary for some of the distributions, we felt it imperative for others,
for example the Gucumatz. A list
of selected percentage points for standardized versions has been provided by ROGERS and TUKEY. unknown parameters, Tukey's percentile to our
X
through
T=a+bX.
In view of the T
So, we estimate
is related a
and
b
using two percentile points. Additional details are as follows: The
C, G
and
L
are well known distribu-
tions and provide no problem. Distributions for the probability mixtures are found by merely mUltiplying the component distributions by their respective -
178 -
probabilities,
for example F S / 4 (t)=0.25 FS(t)+ The Gucumatz distribution is given by
+0.75 FG(t).
if
t<-I,
if
-I:S;
if
t
t
<
I,
~
+pl(F (t)-F (I» C C where FGUCU
and
P2
are subject to the restraint that
must be statistical distribution function.
After
Fx
has been found,
20-th percentiles denoted by ed.
its upper 40-th and
x.4
and
x.2
are comput-
These are put into the linear relationship with the
corresponding percentiles listed by Rogers and Tukey denoted by
t.4
t. 2 , yielding
and
and
From these two equations, found.
Now setting
version of
X.
Now,
the constants
a
and
bare
T=a+bX, we have our standardized given the
FT(t)=Fx(t~a), the
remaining percentiles are easily checked. An example might explain this procedure more clearly. FS
is found using the standard statistical techniques
as: x
FS(X)
=/ ~(e
2
2
0.5
for
xf.O
for
x=O.
Using simple iterative techniques, x. 4 and x.2 are found to be 0.512 and 1.946 respectively. Hence, we have -
179 -
0.512 giving us
and
a+0.256b
and
a=O
b=2.
1.946
a+0.973b,
To check the remaining percen-
tiles, we use t2 FT (t)
Note that,
_ = 2 ( e -8 -1 t
for the GUCU
) +F ( -t
CUO) .
)
G 2
distribution, PI
are
and
found by using the equations
0.8. A comparison of percentage points is given in Table
2.
If a difference occurs, Tukey's value is listed below
in parantheses. This table also provides the reader with a feel for the various distributions. TABLE 2.
DiS~ribu-1
Percentage points for the distributions
40%
20%
10%
5%
2%
1%
0.1%
G
0.253
0.842
1.28
1.64
2.05
2.33
3.09
L
0.254
0.869
1. 38
1. 85
2.44
2.88
4.33
t10n
(1.84) Q
0.254
0.881
1.45
2.10
3.33
4.71
14.9
(4.74) IOG/4
0.255
0.927
2.14
6.52
10.9
13.6
20.6 (20.8)
S/4
0.254
0.867
1.40
2.04
4.36
8.72
87
(R.73)
QS
0.255
0.920
1. 67
2.91
S
0,256
0.973
1.99
3.99
(2.00)
6.48 9.97
12.3
117
19.9
199
Distribu40%
20%
0.256
1.021
tion GUCU
\0%
2%
5%
2.26
4.62
1%
11.6
23.3
(4.61 ) 0.255
3S/4
0.910
1. 75
0.259
1.098
4.62
2.46
233
(23.7) (232) 11.8
23.7
(5.20) C
0.1%
237
(23.2)
5.04
12.7
The differences are extremely minor.
25.4
254
So, we may proceed
to the additional studies involving Monte Carlo methods. Generation of random samples now poses little problem.
Wherever the
through simple forms
v,
wand GUCU,
X-distribution is obtained for
wand
and then divide.
V,
For others,
generating a uniform number -I
through 3.
FX
(y)
we first
generate
such as
and
L
and obtaining
Y
X
proved easier.
BASIS OF SAMPLE CLASSIFICATION
Now,
given a sample, we need a method to determine
from which of the class members it has been drawn. Focussing on the normal as a parent distribution,
a well
known characterization is the independence of the sample mean and variance.
McDONALD and KATTI [2J start their
work by investigating this •.. , x n
of even size
n,
fact.
For a sample
xI ,x 2 '.
their preliminary scatter dia-
grams utilize the independence of the sum and the absolute difference between
and
x.
~
xi +1
for
i = I ,? , .
. ,n-I .
The rationale for this stems from the fact sample, no matter what size, variance.
Hence,
if we plot
have only one point.
that a
has only one mean and one 2 x against S , we would
With this
sole point,
the graph
cannot shed much light on the independence of these two 1 81
variables.
Subsequently, an overall sample of size
should be split into m of just one point.
subsamples,
Of course
giving us
n
instead
m
should be as large as
m
possible forcing the size of the subsamples to be as small as possible. 5
2
At least two points are needed to find
the smallest size for the subsamples is
therefore,
two. For our subsample of size two,
and
x.
~
2 s.
using
1 -2(x.-x· ~
~
which are independent if and only if
~+
and
x.
~
2
l)
and
x.
~
are
are independent identically distributed normal observations.
Hence the variables
and
( 1)
z.
~
x.
which are one-to-one functions of
2 si'
and
~
are also
independent. Since our graph tion,
LS
designed to distinguish distribu-
regardless of location or scale parameters,
tion must be given to the affect on by a linear transformation of =
c+dx i + l •
x'+x' i i+1
Define
and
y1
x.
and
~
Let z1
and
y.
x
=
~
~
~
and
c+dx.
~
similar to
!x~-x~ I!' Then, ~ ~+
attenmade
z.
(I)
y ~
by
2c+ dY i
~
z1 = dlzil. In view of this,
the origin of the scatter diagram
(m ,m) where m and Y z Y tively the sample medians of the set of
is taken to be
m
z
Y's
are respecand
z ' s.
The scale of the graph is set so that
the difference
between the largest and the smallest
z
is fixed as
so many units and the same about the scale on this rule,
y.
\-lith
the graphs focus on trends without regard to
location and scale.
182
-
4. PRELIMINARY EMPIRICAL RESULTS Using computer techniques, ten random samples, each with
n=40, were generated from the distributions men-
tioned in Section 2. Each sample was then split into twenty pairs, a graph was made of versus
Z,
Y,
the pair sums,
the absolute pair differences. Thus, we had
ten graphs from each distribution. To set up an order of presentation, we arranged the distributions in the increasing order of their 1% points for the standardized versions. The order carne out: G, L, Q,
IOG/4, S/4, QS, S, GUCU, 3S/4, C. For a purposes of comparison, three graphs were
chosen from each distribution. One considered "typical", one most like the preceding distribution, and one most like the next. To determine which of the ten graphs would be selected for these, a point system based on empirical criteria was employed. Basically, these criteria dealt with outliers, missing points, and the spread of high and low points. For each criterion, a point system awarded -2, -I, 0,
I, or 2 points to a
graph depending on the number of graphs from the distribution which exhibited said criterion. For the purposes of presentation, only the typical graphs are supplied as Graphs 1-10. The straight lines are
y=m
y
and
z=m, circled points depict repeated z
values, and the number of repetitions is given. Several observations are rather striking. (I) The normal and logistic both lack tails (extreme points).
See Graphs I and 2.
(2) Tails definitely begin to appear in Graph 3 and are prominent in Graphs 5-10. (3) Domination by one tail begins in Graph 4 and is definite in Graphs 5-10. -
183 -
y
Graph 1. Plot of
Y-vs-Z
for the normal distribution
Z
Y
Graph 2. Plot of
Y-vs-Z
for the logistic distribution
-
184 -
z
Graph 3. Plot of
Y-vs-Z
for the
Q
Z
y
Graph 4. Plot of
Y-vs-Z
-
185 -
for the IOG/4
z
y
Graph 5. Plot of
Y-vs-Z
for the
S/4
Z
Y
Graph 6. Plot of
Y-vs-Z
for the
QS
Z
Y
Graph 7. Plot of -
Y-vs-Z 186 -
for the
S
z
Y
Graph 8.
Plot of
Y-vs-Z
for the GUCU
Z
Y
Graph 9.
Plot of
Y-vs-Z
for the
3S/4
Z
Y
Graph 10.
Plot of
Y-vs-Z -
187
-
for the Cauchy
(4) Repeated points is an obvious charactaristic in Graphs 6-9, but is much less apparent in Graph 10.
These observations are made to show trends in the graphs going from the normal-tailed to the long-tailed distributions. Although we have not used the graphs presented by Dr.
Tukey we beleive that
the spirit of our study is in
line with his book Exploratory Data Analysis. We quote from his book, the
'best'
"We do not guarantee to introduce you to
tools,
particularly since we are not sure
that there can be unique bests." Our graphs are relatively simple to construct and exhibit noticeable trends as the distributions become longer tailed.
5. THE STATISTIC The graphs presented in the last section were provided to give a feel for how the relationship between and
z
changes for the various
the variables
Y
distributions.
A quick test can now be performed.
First,
of size n, to find the x l ,x 2 , ••• ,x n Tukey distribution that best approximates the parent n consecutive pairs, population, split the sample into given a sample
(x.,x. ~
~+
I)
Section 3
(i=I,3, ••. ,n-I). Find (y.,z.) ~ ~ n y-z and plot the pairs. The
using
(2)
in
final deci-
sion as to which distribution appears to fit
the data
the best may be based on some point system or on intuitive understanding of the graphs. be that,
Our suggestion would
if one is interested in" such a test, many
graphs should be made from samples from known distributions to gain insight in their intrinsic differences. Being a quick test, with this procedure.
there are obvious difficulties
However,
there are also notable
188 -
strengths. First, the graphs do show obvious trends. Second,
the statistician's personal judgement and exper-
ience are utilized. This judgement should be argumented by a thorough understanding of the
y-z
graph by
constructing these eraphs for distributions of particular interest. To quote Dr. Tukey again,
"exploratory data
analysis is detective work ... ". So,
for this large class, we have a general eye-
-ball scheme. Restricting our attention to only three alternative, normal-tailed,
long-tailed, or skewed to
the right, we have a more organized study which shows much power.
This indicates that the quick test can have
much value. McDONALD and KATT!, referenced earlier, based their test for normality on the independence of the variables Y
and
z.
Specifically,
they employed a runs test and
provided a scheme to accept or reject normality. We propose a test that not only distinguishes between normality and non-normality, but also indicates whether a non-normal dis:ribution is either long-tailed or skewed. Thus, we are really providing for a test of 3 hypotheses: H):
the data is distributed normally,
H2 :
the data is distributed as a chi-square with 2 degrees of freedom,
H3:
the data is distributed as a Cauchy.
For this 3-way test,
our suggestion is to consider
a second order, linear regression model:
z
(2) If
Y
and
coefficients
2
ct+I3Y+yY +E.
z
are independent, then the regression and
y
are both equal to zero. An
F-like ratio that is, dividing the regression sum of )89 -
squares due to the individual coefficients by the error sum of squares,
is a natural statistic for testing that
both coefficients are equal to zero.
Since the basic
assumptions of normal regression analysis are not met, our ratios cannot be compared to the percentage points of the
distribution.
F
In a later section, an empirical
study provides regions of acceptance for the various hypotheses yielding power of roughly 80%. 6.
PROCEDURE FOR THE 3-WAY TEST
Given a sample
of size
xI ,x 2 ' ••• ,x n
n, we would
like to find two dimensional vector test statistic
(R I ,R 2 ). The basic idea is to rewrite equation (2) with the second degree orthogonal polynomial
where
=
~I (y)
y+d
and
are the orthogonal polynomials and
a,b
and
care
found using the least squares criterion. A fundamental alteration from our quick test of Section 5 is that the
(y, z)
pairs are now found using
all combinations of data points, not merely the consec-
utive pairs.
This is done to extract maximal information
from the sample. For this reason, equations changed to (3)
where
y .. = x .+X. ~]
S i
~]
<
j
and
z .. ~]
S n.
-
190 -
Ix .-x.1 ] ~
(I)
are
To find the constants
(~)
summations over all the tions.
a,b,c,d,e l
possible
and
e2
requires combina-
(y,z)
Extensive simplification of these sums can be
found when the data is listed in descending order, x(I)'
A list of simplifications needed
x(2) , ••• ,x(n)'
to find
as given below:
(R 1 ,R 2 )
(a)
'Ly ..
(b)
'Ly ..
(c)
'Ly ..
(n-4)'Lx(i)+3('Lx(i»
(d)
4 'L Y ..
(n-8)'Lx(i)+4('Lx(i»
(e)
'Lz ..
'L(n-2i+l)x(i)
(f)
2 'Lz ..
n'Lx(i)-('Lx(i»
(g)
'Lz .. y ..
'L(n-2i+1
(h)
2 'Lz .. Y ..
'L (n - 2 i + 1 ) x
(n-l)Lx(i)
~J
2
2
3
~J
~J
~J
3
2
4
3
2
1.J
~J
2
(n-2)'Lx(i)+('Lx(i»
~J
~J
say
~J
~.J
('Lx(i»
2
('Lx(i»+3('Lx (i ) )
2
2
)x~i)
~ i) + .~'L. x (i ) x (j)
Now we are ready to list the steps
(x (i ) - x (j ) ) .
J
~
~n
finding the test
statistic (I)
Take a sample of size
n
and list its members
in decending order. (2)
Find
d,e l and e2 using the properties of orthogonal polynomials, resulting with d
=
-'Ly/nc 2 ,
and (3)
Find
e2
=
e l
del-'Ly
a,b,
and
= 2
«'Ly /nc 2 ,
c
2
) (Ly)/nc 2 -'Ly
where
nC 2
3
=
2 )/('Ly +d'Ly) n
(2)'
using ordinary least squares. 2
a = 'Lz/nc 2 , b = ('Lzy+d'Lz)/('Ly +d'Llj),
and
c =('L zy 2+ el 'LZY +e2'Lz)/('Ly4+el'Ly3+e2'Ly)2
-
191
-
(4) Compute the sum of squares due to the coefficients and error through the use of the formulae SSa= aEz, SSb
=
b(Ezy+dEz),
2
SSe= e(Ezy +e I Ezy+e 2 Ez),
and
SS = Ez 2 -(SS +SSb+SS ). e
(5)
a
e
The statistic and where
6 (w)=
6(w)
I
R2
=
6(e)(SS )/SS e
e
,
is defined by
-I
if
w
0
if
w
if
w
<
0 0
> o.
The final decision as to which of the three hypotheses in Section 5 to choose is based on empirical, Monte Carlo results presented in Section 8. 7.
INVARIANCE OF
One last consideration must be noted. To deal with the composite hypothesis of normality with unspecified mean and variance, our statistic should be linearly invariant. To check if it is so, consider the linear transformation of our original data with
m~O.
~J
t
m(x i + "2)'
Now, let and
where
x! .
z
~
.
~J
1m! z .. , ~J
and
z ..
are defined as in (3) in Section
Looking at
RI ,
the ratio of the sums of squares,
~J
6. sSb/SSe' is known to be linearly invariant due to its -
192 -
similarity to an
statistic. Therefore, it suffices
F
where
6 (b')
to consider
b
is
I
b's equivalent when
working with the transformed data discussion,
let the symbol "
,
X'.
Throughout the
" indicate the equivalent
constant or variable when working with the transformed data. d
It is easily seen that
b
m(d-t). Now writing
I
Ez(y+d) Ey(y+d)
we see that
b '
Im!mEz(y+d)/mZE(y+t)(y+d)
E(y+d)
since
6 (b' )
o.
I
!m!b/m,
Hence
-6(b)
if
m
<
0
6 (b)
it
m
>
O.
is only invariant up to its sign.
Such depen-
dence only causes problems with skewed distributions. Hence,
mu~t
the skewed alternative
tion of the tail.
indicate the direc-
In our work we assumed that the tail
was to the right. Turning to
RZ '
again it suffices to consider only
6(c'). After a little algebra, it is seen that e ' 1
and
e
I
Z
Now, '''riting
-
193 -
22 42 22 11 m m Ez(y +e l y+e 2 )/m E(y +2ty+t )(y +e l y+e 2 )
c'
I
I 2 Im,c/m ,
since
=
is found to be linearly invariant.
6 (c)
8. EMPIRICAL RESULTS USING To determine our various acceptance regions, we generated 500 samples of size
from each of the
n=20
parent distributions normal, chi-square with two degrees of freedom,
and Cauchy. For each distribution, the 500
pairs were found and graphed using
APL. Graphs
11-13 are the overall graphs for the three distributions. To indicate the number of repeated points at a position, an intricate system was employed. A point which repeated 1-9 times was indicated by that number. A point which repeated 10-35 times was indicated by the letters with
A
indicating 10, B I I , and so on up to
ating 35 repetitions. Points repeating 36 to
Z
A-Z indic-
135 times
were represented by an asterisk and those repeating more than 135 times by "+".
Since
.'.
and
+
do not
specify the exact number of points, the horizontal row sums are provided in parentheses in the right margin for partial guidance. We divided the
space into three regions
so as to get a power of approximately 80% for each of the three alternatives. To draw these regions, we redrew graphs
I I,
12 and 13 with a wide variety of
magnifications of different regions. Graph 14 gives our final acceptance regions for the three alternatives. Next,
10,000 new samples with
n=20
were generated
from each parent distribution and classified according -
194 -
R2
0.25
1
1
0.00
1 1 1
0.25
0.50
1 1 12 1 1 1 12 11 2 26K521 1 1 11 11147FQ*H5725131 212 1 1 13 1325A*77431 1 122 AK351 1 21144314411 111662 11 1 1 11731 1 15 22 1 1 2 1 12 1 1 11 2 1 1 1 1
(40) (203) (87) (46) (43) (20) (15)
R)
0.75 1.0
1.5
Graph
II.
R I -vs-R 2
0.5
0.0
0.5
1.0
for the normal distribution
0.25
r R:,
0.00
0.25
0.50
0.75
1. 00
1 1
1 11 1 1 11 1 1 11 1 12 11 2 2 1 1 311 1 1 ( 21) 1 1 1333398496397484 3 221 3 1111 1 1141 1 (1.20) 1 2 1 228347474324223411 1 111 1 11 (70) 1335134 83 123 1112 (45) 1 1 1 156525 15123121 1 2 1 2 (48) 1 1 (26) 11 1 ~ 22212111 211112 111 415 22212221 2 1 (27) (19) 2 12 211121 2 1 11 (16) 11 1 2221111 11 1 (10) 1112 131 (13) 2 1 l3 11 1 12 1 (14) 113 1 1 2 1 1 111 1 1 1 11 1 12 1 1 1 1 12 1 2 1 111 1 1 11 1 1 1 1 1 1 1 1 RI 1 1 1 1 I 5 0 4 2 3 1
for the chi-square distribution Graph 12. R 1 -VS-R 2 with 2 degrees of freedom
-
195
-
7.5 1 1 1 3
5.0
6
3 1
2.5
4 9
1
2A
0.0
1 1
1
1
(l3)
L2
(23)
Xl 1
(34) (49)
* 1 11*212 2* 1 A 2
(117) (221) (10)
1
5.0 .25000
10000
15000
20000
o
5000
5000
for the Cauchy distribution
R2 CAUCHY CAUCHY
-.2
NORMA~·l .25
I
CHI .25 CHI
R) 5.5
NORMAL -0.3
CAUCHY
CAUCHY
-1.0 CAUCHY
Graph 14. Acceptance regions -
196 -
to Graph 14. Table 3 records the results of this power study. In this table,
the strength of our test statistic is
clearly exhibited.
For samples with
n=20, it indicates
that roughly 83% of those samples taken from any normal distribution will be classified as normal, 88% of those taken from a chi-square with 2 degrees of freedom will be correctly classified, and the proportion of correct classifications for samples from the Cauchy distribution
(R 1 ,R 2 ), and therefore its basis is shown to be an effective discriminator of
is roughly 78%. Hence (y,z),
distribution. Table 3. Monte Carlo Power Study Proportion of the time accepted
Distribution normal
chi-square
Cauchy
normal
0.8344
0.0753
0.0875
chi
0.0752
0.8869
0.1237
Cauchy
0.0904
0.0378
0.7888
9. CONCLUSIONS Two major results are presented in this paper. First, that the variables
Y
and
Z
defined in equa-
tion (I) can distinguish symmetric distributions as to the length of their tails to a reasonable extent. And, if
Y
and
Z
are extended as in equation (3),
give rise to a test statistic
they
(R 1 ,R 2 ). This statistic
is shown to have approximately 80% power when applied to the normal, chi-square with two degrees of freedom (tail to the right), and the Cauchy distributions.
-
197 -
REFERENCES [)J
K.O. Bowman - L.R.
Shenton, Omnibus test contours
for departures from normality based on b 2 , Biometrika, 62(1975), [2J
L. McDonald -
~
and
243-250.
S.K. Katti, Test for normality using
a characterization of the normal distribution, unpublished, [3J
1974.
J.K. Ord, On a system of discrete distributions, Biometrika, 54(1967), 649-656.
[4J
J.K. Ord, Graphical methods for a class of discrete distributions, JRSS,
[5J
E.S.
Pearson - N.W.
130(1967), 232-238. Please, Relation between the
shape of the population distribution and the robustness of four simple test statistics, Biometrika, 62(1975), [6J
C.A. Rogers - J.W.
223-241.
Tukey, Understanding some long-
-tailed symmetrical distributions, Statistica Neerlandica,
26(1972), 211-226.
S. K. Kat t i and J.
Stith
2 2 2 lIa t h • Sci. Bid g . University of lIissouri Columbia, 110 65211, USA
-
198 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
A CHARACTERIZATION OF GENERALIZED NORMAL AND GAMMA DISTRIBUTIONS K. LAJK6
INTRODUCTION Well-known characterizations of the family of normal and gamma distributions are the following: The independent random variables normal distributions if and only if
X
and
y
X+Y
and
X-Yare
have
independent (see [14J). The independent random variables X X Y
and
Yare gamma distributed if and only if
X+Y
and
are independent (see [15J). Further characterizations
were given among others by M.V. TAMHANKAR [20J, P. FLUSSER [5J, I. AITCHISON [IJ, E. ROSLONEK [18J, I. KOTLARSKI [9J, R. KR6LIKOWSKA [10J and H. JAKUSZENKOV [8J. Our main result is a characterization of the undermentioned generalized normal and gamma distributions: Let
X
be a continuous random variable. We say
that: A)
X
has a generalized normal distribution if its
probability density function is
-
199 -
__ 1- ( S (x) -ll ) 2 2 -1-ISI(x)le 20 for
(I)
f(x)
S
where
'l~"a' (a,b)
R
onto
oER,,{O}
and
B)
otherwise,
is a continuously differentiable function from
the interval xE(a,b)
xE(a,b)
such that
is an arbitrary
S I (x)
0 * constant.
for
has a generalized gamma distribution if its
X
probability density function is
f(x)
(2)
otherwise, where
a
is a continuously differentiable function from
the interval
(a,b)
xE(a,b)
A,pER+
I.
and
onto
R+
such that
al(x)
*0
for
are arbitrary constants.
PRELIMINARY RESULTS
Our characterizations are based on the following results THEOREM 1.1
(see [3J,
measurable functions
[12J). Suppose that the
f,g,h,k:
R -
R
satisfy the func-
tional equation (3)
where
f(x)g(y) a,b,c,dER O
6 = ad-bc
* 0,
h(ax+by)k(cx+dy)
=
R"{O}
are arbitrary constants with
and there exist sets
Lebesgue-measure such that yET 2 • Then
f,g,h,k
(x,yER),
f(x)g(y)
have the forms -
200 -
T I ,T 2 CR of positive 0 for all xET I ,
*
2
(I•I)
f(x)
cxlexp[ alx+blx ]
( I .2)
g(x)
cx 2 exp[a 2 x-
( I .3)
h (x)
B l cx l cx 2 exp
( I .4)
k(x)
B2 cx l cx 2 exp
where
(xER ),
bd 2 blx ] ac
[ a l a-a 2 e ~
x + -a b x 2 a I
[a 2 a-a l b ~
cx.,B.ER ~
(xER) ,
~
- £.c
1
b x 2] I
(i=1,2)
(xER ) ,
(xER) , are arbitrary
constants such that ( I .5)
I.
THEOREM 1.2
(see [4J, [IIJ). Suppose that the
measurable functions
f,g,P,q:R+ -
R
satisfy the func-
tional equation (4 )
f(x)g(y)
p (x+y) q(~) y
T I ,T 2 CR+ of positive Lebesguef(x)g(y) 0 for all xET I , yET 2 .
and there exist sets measure such that Then
f,g,p,q
*
have the forms
(I. 6)
f(x)
A
exp[ ax+blnx]
(1.7)
g(x)
B
exp[ ax+ (c-b) lnx]
( I .8)
p (x)
C exp[ ax+clnx]
(I. 9)
q(x)
D exp[blnx-cln(x+I)]
- 201 -
(XER+), (XER+), (XER+), (XER+),
where
a,b,cER
and
A,B,C,DER O
are arbitrary constants
with (1.10)
AB
CD.
THEOREM 1.3 (see e.g. [6J). Let
an
X
=
(XI, .•. ,Xn ) n-dimensional continuous random variable with a
probability density function
y.~
which is zero outside
n x CRn • Let
of a region (5 )
f,
be
=
Ql.(xl,···,x ~ n)
(i=I, •.. ,n)
be a one-to-one transformation defined in the region S1
x
and denote
(6 )
x.~
=
~'(YI' ~
.. . ,y n )
(i=I, .•. ,n)
its inverse transformation, which is defined in a region S1 CRn.
!l. If the Jacobian
J(YI' ... 'Yn)
a(xl,···,x n ) a(yl,···,y)
is continuous, and does not change signs in the
n
exists,
n then
}L
n-dimensional random variable y
••• ,Ql n (xI'···'x» n is continuous with a probability density function
g
such that f(~I(YI,···,yn)'···'~n(YI,···,yn»x
xIJ(YI' ... 'Y n )!
(7)
o -
for
otherwise. 202 -
}LEn}L'
2. CHARACTERIZATION OF GENERALIZED NORMAL DISTRIBUTION Let
be independent continuous random
XI""'X n
variables and define the random variables
YI""'Yn
by
n
F I[
L
»),
f\(X i
i=1
(2. I) F.[8 1 (X I )-8 (x ), •.• ,8 I(X 1)-8 (x») ~ n n nnn n
Y.
~
(i=2, ••• ,n),
where the transformation n
FI[ (2. I ' )
~
»),
8 i (x i
i=1
F.[8 1 (x l )-8 (x ), .•• ,8 I(x 1)-8 (x») ~ n n nnn n (i=2, •.. ,n)
~ CRn
maps the region
onto
x
nx. ....
~
(i=I, ..• ,n);
~
.... n
R
FI
(where
R
and satisfies the
Y
with-functions
conditions of Theorem 1.3 8.
n CRn
YI
n x ... xn xI
xn
Rn - I
; F.
~
.... n
Yi
n , n x ... xn x
YI
(i=2, • •• ,n)
Yn
n ). The inverse JL
transformation are defined by x.
~
-I[
B.
=
~
(2. 2)
] F -I (Y 1 )- n-I ~ G.(y 2 , ••• ,y ) i= 1 ~ n G.(y 2 , ••• ,y)+ ~ n n n-I
x =13 n
-IIF~I(YI)-.EIG.(Y2""'Y n )] ~
~
n
n
-
203 -
(i=I, ••• ,n-l)
n , where the functions
in the region _ R
G.: ~
JL are the inverses of functions
n
Y2
x ... xn
Yn
(i=2, ••• ,n).
y.~ = F.(zl'···'z ~ n- I)
The Jacobian of the transformation (2.1 ') is
J(xI,···,x)
=
n
(2.3) XH( 8 I (x I ) -
8 (x ), ... , 8 n
n
xEn , where x
for all
Fi[ i=1 ~
H:
8.(X.)]x
n-
~
~
I (x
n I ) - 8 (x )] IT 8 ~ (x . ) nn n i=1 ~ ~
Rn - I -
R
is a certain func-
tion. From (2.3) we get
x H[GI(Y2' ••• 'Yn), ••• ,Gn-I(Y2' ••• 'Yn)]x
(2.4)
x 8'
n
[8
n-I
-I n
[ -I
x IT 8! 8. i=1
(
~
~
-I FI
n-I
(YI)-·~IG.(Y2'···'Y) ~-
n
~
(G.(Y 2 , ••• ,y )+ ~
n
-I 'I
1 )Jx
n-I
(YI)-j~IG~(Y2'···'Yn)
l]
n
for the Jacobian of the inverse transformation (2.2)
in
n . JL
We prove that the independent random variables have generalized normal distributions if and
XI' ••. 'X n
only if the random variables
YI
(where
are independent.
Y.
~
denoted by (2. I»
and
(y 2 , ••• ,Yn )
Our proof based on Theorems 1.1 and 1.3. We need the following
- 204 -
LEMMA 2. I. Suppose that the functions g: R" Rand
(i=I, ••• ,n),
f.
~
G: Rn - I .. R
:
R .. R
satisfy the
functional equation n
n g( ~
II f.(x.)
(2.5)
~
i= I
x.)G(xl-x
i= I
~
and there exists sets
A.CR
n
, ••••
x
I-x)
nn «xl·····xn)ER)
~
(i=I ••••• n)
~
n
of positive
Lebesgue-measure such that n fi(x i ) ~ 0
II i=1
(X.EA.; i=I ••••• n). ~
~
Then n
II
f.(x.)g(u l )G(u 2 ••••• u) '" 0 ~ ~ n
i=1
x .• u . ER ~
PROOF.
(2.5)
for all
(i = I ••••• n) •
~
implies that
n g(
for all
~
x.)G(xl-x ~
i=1
x.EA. ~
~
n
, ... ,x
(i=I, ... ,n),
n-
I-x)
n
'" 0
that is
(2.6)
for n u E
~
I
i= I
(U 2 ,U 3 , ..• ,U)E
n
x
U
n
EA
[(AI-x )x •. • X(A
n
n
Substituting
-
205 -
n-
I-x)]
n
n U
I
=
(i=1,2, ••• ,n-I),
~
i=1
we get from (2.5) the equation n-I n-I n uI-';~1 u. I uI-·~lu. I IT ( ~ ~+)f ( ~= ~+) i=1 fi u i + 1 + n n n (2.7)
n IT f.(x.)*O i=1 ~ ~
which together with (2.6) implies that
x .EA ~
n-I [--(A.-x )+
u
~
~
n
x EA n n
n n
EA'
n
x
u
n
~
c= .
EA
n
n-I A.- ~ (A .-x i=1 ~ j=1 ] n
!( ~ n
»)
]
j*i
and x
n
~
if
n-I 1
A.~
.
~
~=
n
1
(A.-x)
n
~
n
).
Since n-I ~
A!::> .1l. + ~
~
n j=1
(A.-A.) ] ]
(i=I, ••• ,n-l)
j*i
and n-I A'::> A
n
n
+ n
~
j= 1
(A. -A .) , ] ]
we have
(?.8)
n IT f.(x.) ~ ~ i=I
*0
if n-I
x.E A. + n ~ ~
~
j= 1
(A. -A .) ] ]
- 206 -
(i=I, ••• ,n-l)
and n-I
:E (A. -A .) •
+
n j=1
]
]
By a theorem of Steinhaus (see [7J) the sets
A.-A. ~
~
(i=I, ••• ,n-l) tervals
contain intervals. Thus there exist in000 I. [a.,b.]CR (i=l, ••• ,n) such that
(2.9)
n
~
~
~
n
'*
f.(x.) ~
i=l
~
o
if
0,
Repeating this argument
(i=I, ••• ,.n).
x.EI. ~
~
o
(used
instead of
I.
~
A.), we ~
have n
n
(2.10)
'*
f. (x.) ~
i=l
~
0
if n-I
1
x .EI. = I~ + ~ ~ ~
o 0 :E (I.-I.) n j=1 ] ]
(i=I, ••• ,n-l)
j*i
and x
n
EI I
= I
n
O
n
n-I
+
0
0
]
]
:E (I .-I.).
n j=1
It is easy to see that the sets (2.10)
I1
(i=l, ••• ,n) in
are the intervals
(2. I 1 )
+~
n-I
+
n
1:
j=1 j*i
o
0
]
]
(a.-b.),
b~ ~
+ n
n~1 (b~-a~) 1 ] ]
j=1
j*i (i=1.2, . . . ,n-l)
-
207 -
(2. I I ) n-I ~
0
0
bO
~
~
n
(a.-b.),
n i=1
0 0]
n-I
+ ~ (b. -a .) n i=1 ~ ~
•
Thus n (2. 12)
n
f.(x.) ~ ~
i=1
'" 0,
if
(i=I, ••• ,n).
By induction, we get a sequence (i=I, •••
,n;
k=0,1,2, •••
with property
)
n-I
0
0
j= 1
]
]
:E (a .-b .), b Oo + k-
j
~
n
°1
n1 0 ~ (b.-a.) j=1
]
]
",i
(2. 13)
(i=I, .•• ,n-l)
~[
0
0
bO
~
~
k + -
n
n
'" 0,
if
Ok
n-I
n
i=1
an +
~
(a.-b.),
n-I ~
i=1
0
0
~
~
(b.-a.)
J
and n
n i=1
f
i
(x.) ~
(i=I, •••
-
208 -
,n; k=0,1,2, ••• ).
(k)
From (2.13) one can see that as
k
-
a.
_ ""
-00
~
therefore
"",
n
n
i=1 This and
f.(x.) ~ ~
(2.5)
if
'" 0,
x .ER ~
(i=I, ... ,n).
gives that
u .ER ~
(i=I, ... ,n),
which completes the proof of Lemma 2.1. Now we can easily prove
Let
THEOREM I.
XI""'X n
be continuous and
independent random variables with densities (i=I, ... ,n). Let
YI""'Y n
fi
nx
R
be continuous random var-
iables defined by the one-to-one transformation which maps the region
R -
n. y
onto the region
(2. I)
Further
suppose that the Jacobian of the inverse transformation
(2.2) exists, is continuous and does not change signs in
ny .
Then
XI'"
.,Xn
have generalized normal distribu-
tIons with densities
(B.(x.)-fJ.) ~ ~ ~
(2. 14)
f.(x.) ~ ~
fJ.ER ~
} (x.Er/ ~
=1 o
(0,
2
(x .ER'\n ~
(i=I, ... ,n)
x.
)
are arbitrary constants,
the
B; : n x.
n
R ) i f and only i f the random variables
x.~
and
onto
~
(i=I, .•. ,n)
maps the intervals
~
are independent.
-
209 -
)
~
~
functions
R
x.
PROOF.
••. ,Yn )
YI and (Y 2 , ••• then by Theorem 1.3, we have
If the random variables
are independent,
(2.15 )
g: R -
where
G : R
Rand
density functions of
YI
n-I
-
R
are the probability
(y 2 , ••• ,Yn ),
and
Hence by the help of transformation
respectively.
(2. I) we get
n g[ F I (~ i=1
( 2. I 6 )
X
XG[ F 2 (v I ' ••• , v n _ I ) , ••• , F n (v I ' ••• , v n _ I )] X
X1Fi[ i=1 ~ for all
Bi (x i ) ) ]
B.(X.)]H(VI, ••• ,v _I) ~
~
n
(xl' . . . 'x )En , where n
(i=I,2, •• • ,n-I).
x
~
~
v. = B.(x.)-B
n~
n
Furthermore
~ B~(x.)1
i=1
~~
f.(x.)
i=1
~
~
= 0,
(x )
nn
if
xERn-n x
By the substitutions
(2. 17)
Bi (xi)
(i=I, ... ,n; x .En
t.
~
~
-I
(2. 18)
(2. 19)
f. (t .) ~
~
g(z I)
fir Bi
(t i )]
I Bl[ B~I (t i g[ F I (z I )]
)]
210
)
-
,
~
(i=I, •••
I
I F i (z I ) I
-
x.
,n;
(zIER),
tiER) ,
G (z 2' ••• , Z n) =G[ F 2 (z 2' ••
0
,
Z
n) , ••
0
,
F n (z 2 ' ••• ,
Z
n)] X
(2.20)
(2.16)
(2.21)
goes over into the functional equation n n IIf.(t.)=g( ~ t.)G(tl-t , ••• ,t I-t) i=1 ~ ~ i=1 ~ n nn n ( (t 1 ' ••• , t n) ER )
for the functions
f i ,
g :
are density functions, where zero. f.
~
R -
G :
R,
Rn - I -
R.
fi,g,G
thus they cannot be almost every-
By (2.18) the same applies to the functions
(i=I,,,.,n).
Using Lemma 2.1, it follows that n
II
for all
'# 0
f.(t.)g(u 1 )G(u 2 , . " , u ) ~ ~ n
i=1
u.ER
t ., ~
(i=I, ••• ,n).
~
Now, let
be fixed and
j'#n
t.=t ~
n
if
Hj.
Then
we get from (2.21) that n f.(t.) II f.(t.)= ] ] i=1 ~ ~ Hj
=g(t.+(n-I)t ]
for all (2.22)
for all (2.23)
t.,
]
t
n
ER.
]
)G(O, . . . ,O,t.-t ]
n
,0, ... ,0)
This implies the functional equation
f . ( t . ) f (t ) n n J ] t
n
g(t.+(n-I)t ]
n
)G(t.-t ) ] n
.,t ER, where n
f(t)= n n
n IIf.(t) i=1 ~ n i'#j -
(t ER),
n
211
-
(2.24)
G'(O, .•• ,0,
G(z)o
z
(zER) •
,0, ... ,0)
j
The functional equation (2.22) is a special case of (3) with t
a=l,
b=n-I, c=l, d=-I.
Thus the functions
ER.
0' 1 , g,
f
n J n conditions of Theorem 1.1 since f
g, G
1 n (t n )¢O
Further
o
,
~
f.
are measurable. Therefore
G
g,
G
f., f
and
and
J
for all
satisfy the J
n
,
1n
are of the
:s; j
<
forms (2.25)
f . (t
(2.26)
1
J
n
a.t .) J
(t )
n
=
J
(t ER • J
J
0
a(j)t +(n-l)b.t 2 a(J)e n n J n n
n)
o
n
J
(t ER), n
are arbitrary con-
a .,a(j),b oER
and
n
J
o+b.t~ J
J
a 0' a(j)
where
J
= a .e
J
stants. From (2.22) using (2.25) and (2.26), we get n-I
n-l
(a.+a(j)- ~ a.)t +(nb o- ~ b.)t 2 J n i=1 ~ n J i=1 ~ n
f (t )= n
-
n
Thus
j¢n.
for any fixed and
-a +a (k) k n
n
i=1
~
n
J
n
a
and
n
a
By
the expressions
depend only on
( 0)
a .+a J J
n
=
a o-a(j)J n
n. Denoting
a.
~
i=1
these by
k
a(j)a. n~1 J
and
a.
b l =b 2 = .•• =b n _ l =b.
a(j)a.=a(k)a
n-I ~
(t ER)
n
respectively we get that
n
2 a.t.+bt. (2.27)
where
f.~
(t .) ~
a.e
~
~
~
~
aiER+, ai,bER
(t .ER, i=I, •.. ,n), ~
(i=I, .•• ,n)
-
212 -
are arbitrary constants.
Then (2.17) and (2.18) gives
(2.28 )
fi
(xi ) --~i 1'( Bi xi )1 e
2 a.B.(x.)+bB.(x.) ~~
~
~
(x.En ~
and
f.(x.) ~
~
x.Efn x.
for
= 0,
The functions
(i = I
~
x.
,
,1• • • ,
n)
are den sit y fun c-
tions thus it is easy to see that
2
jJi
jJi
2"
i=I, ••• ,n),
~
(i=1,2".u,n).
~
f.
~
e
~.
~
o
which together with (2.28) implies
20 2
(2.14).
Thus the if part of our theorem is proved.
XI' ... 'X n be independent random variables having generalized normal distributions (2.14). b) Let now
Using Lemma 1.3
and the Jacobian (2.4) of the
transformation (2.2) we get for the density function of the
(yl, .•• ,Yn )
n-dimensional random variable
expression
e
-20 2
Xe
In-I Xe
n
--2 2<'~ G'(Y2'···'Y )-.~ jJ.) n0
~
=I
~
n
~
=I
~
X
n-l
-2 , . 2: ( G. ( y?, ••• ,y )-jJ.) 2 0 ~ =I ~ n ~
xe
It is easy to see that
-
G
213 -
can be written as
G the
where
(2.29)
Iile
X
U21ta 2 )n-1 1
n-I
n
n-I
1
exp{- 2 a2( ~ Gi (y 2 ,oo.,Y )- ~ ~.)- 2 2 ~ (G·(y 2 ,···,y )-~.) n i= 1 n i= 1 1. a i= 1 1. n 1.
2
}
X ------------------------------------~~------------------H (G 1 (y 2 ' ••• , Yn) , ••• , Gn-I (y 2 ' ••• , Yn))
I
i.e.
Y1
I
and
are independent.
(y 2 , ••• ,Yn )
Thus our theorem is proved.
Remarks 2.1. The formula
g
(2.29) gives the density function
of random variable
defined by
Y1
(2.1)
in the form
for (2.31) for Similarly (2.30)
implies the density function of
(y 2 , ••• ,Yn ). (2.31) and (2.29) shows that Y1 has also a generalized normal distribution (I) with S(y l ) = -I n F 1 (Y I ) ' and parameters ~ = ~ ~. and lila. Further i= 1
if
Si(x i )
=
xi
(i = 1 , ... , n),
l.
F 1 (Y 1 ) = Y I'
we get t hat
the sum of normally distributed random variables is n normally distributed with parameters ~ = ~~. and i=1 1. lila, which is well-known (see e.g. [6J). 2.2. Let all
n=2
and
F.(x)=x, 1.
xER. Then Theorem 2. I.
S.(x)=x (i=I,2) l.
racterization of normal distributions
-
for
eives the well-known cha-
214 -
(see e.g.
[14J).
2.3. If
=
B(x)
log x,
then (I) is the lognormal
distribution and our theorem is a characterization of lognormal distributions. 3. CHARACTERIZATION OF GENERALIZED GAMMA DISTRIBUTIONS Let
XI , .•.
,xn be independent continuous random var-
iables. Let us define the random variables
FI [
(3. I)
~
i= I
~
by
a. (x . )] , ~
~
a
a I (x I) Fi(a (x ) n n
Y.
YI' ••• 'Y n
, ••• ,
n-I
a
n
(x ) n-I) (x )
(i=2, ••• ,n),
n
where the transformation
FI [
(3. I ')
~
i= I
a.
~
(x.)] ~ , ... ,
maps the region
n
C Rn
x
~
n-I a
n
onto
the conditions of Theorem 1.3
a.: n
a
y
n
(x (x
)
n-I)
n
)
C Rn
(i=2, ••• ,n)
and satisfies
with functions
(i=I, .•• ,n);
x.
~
(i=2, ••• ,n)
(where
x ••. x n
x
= n
n x'
The inverse transformation are defined by
-
215 -
G'(Y2""'Y) ~ n n-I
x.
~
~
1+
(3.2)
i= I
)
(i=I, ... ,n-l)
G.(y 2 , ... ,y ) ~ n
-I
x
n-( ____~F~I--(-Y~I-)-------)
n
n
n-I ~
1+
i= I
in the region ••• X
y.
~
n
Yn
-
n , ~
G·(Y 2 ,· . . , Y ) ~ n
= F , ( z l " " ' z n- I) ~
~
Y2
x ...
are the inverses of functions
(i=I, ••• ,n-l)
R
G.:n
where the functions
(i=2, ... ,n).
The Jacobian of the transformation (3.1') is n
Fi[
~ n.(x.)] i= I ~
~
n
i= I
n!(x.) ~
~
nn-I (x )
n
x
n
(3.3)
, ... ,
n
(x ) n-I n-I) n (x )
n
f or a 11
En x.., x
-
h were
n
H·.R n - 1
R
is a certain function.
From (3~3) we get
n-I
(3.4)
X[I+
~
G.(y 2 , ••• ,y)]
i=1 ~
n-I
X
n
XH[ G I (y 2 ' • • • , Y n) , • • • , Gn _ I (y 2 ' • • • , Y n ) ] X
for the Jacobian of the inverse transformation (3.2) in -
216 -
To characterize the generalized gamma distribution (2), by independence of the
and
following
LEMMA 3. I.
Suppose that the functions
(i=I, ... ,n), g:R+ -
G:R~-I - R
and
R
fi:R+ -
R
satisfy the
functional equation n
n
n
(3.5)
~
i=1
XI x. )G(~ x n
g( ~
f. (x .) ~
i=1
T.CR
and there exist sets
~
,
... ,
(i=I, ••• ,n)
x
..E.:..!. ) x
n
of positive
+ n
n f.(x.)#O for all x ~.ET.~ i= I ~ ~ for f.(x.)g(u l )G(u 2 , ••• ,u )#0 ~ ~ n
Lebesgue-measure such that (i=I, ••• ,n). Then all
x. , u .ER ~
~
n
n
i=1
(i=l, ••• ,n).
+
The proof of this lemma is similar to that of Lemma 2.1,
therefore it is omitted.
Then it is easy to prove THEOREM 2. Let XI' ••• 'X n be continuous, independent random variables with densities f.:R - R (i=I, ••• ~
••• ,n). Let
YI' ••• 'Y n be continuous random variables defined by the one-to-one transformation (3.1), mapping the region
n
x
CRn
onto the region
n
CRn. Further
~
suppose that the Jacobian of the inverse transformation (3.2)
exists, is continuous and does not change sign in
n • Then
xl, ••• ,X
n tions with densities !L
have generalized gamma distribu-
-
217 -
Pi ~(
(3.6)
f. (x.) ~
=1 0
P . -I -aa . (x. ) ~ e ~ ~ ,
) Ia ! (x .) i[ a . (x . ) 1
~ ~
Pi
~ ~
x.En ~
x.
~
~
x.Efn ~
x.
~
a, PiER+ (i=I, ..• ,n)
(where
a.
the functions onto
are arbitrary constants,
(i=I, •.• ,n)
~
PROOF.
(Y 2 , ••. ,Y 2 )
x.
~
R+) i f and only i f the random variables
(y 2 , .•• ,Yn )
n
mape the intervals
and
YI
are independent. (a)
If the random variables
and
YI
are independent then, by Theorem 1.3, we
have
(3.7)
(yEn ),
-
where
g:R -
R
and
G:R n - 1
density functions of
YI
-
and
are the probability
R
(y 2 , ••• ,Yn ),
Hence by the help of transformation
n
n
n
a! (x . )
[ n [ n~ a.(x.) 1~=_I g FI ( ~ a.(x.» 1IF; IX i=1 ~ ~ i=1 ~ ~ an (x) 'I~~
f.(x.) = ~
i=1 ~ (3.8)
respectively.
(3.1 ') we get
the functional equation
n
~
n
X G(F 2 (u l , ••• ,u n- I)' •••• Fn (u,I ••••• u n- I»
n
X
a.(x.)
for all
(xl •••• ,x )En
n
x
where -
218 -
u.~
~
a
n
~
(x
n
)
• Further
n
n
= 0,
f.(x.)
i=1
~
~
.xEf nx . -
for
By the substitutions (3.9)
a.(x.) ~
~
t.
=
,n;
(i=l, •••
~
x.En ~
x.
)
~
-I f.[a. (t.)]
f.
~
~
(t .)
~
~
(i=I, ••• , n - I ; t .ER ), ~ +
I a~(a.-I (t.))1
~
~
~
~
(3. 10) [a-I(t
f
f
(3• I2)
n
(t
n
n
)
n
!a'n (a-I n
G ( z 2 ' ••• ,
Z
n)
)] n
(t
tn-I
(t ER
n
» I n n
= G[ F 2 (z 2 ' ••• ,
«
Z
+
),
n) , ..• , F n( z 2 ' ••• ,
n-I
z2""'z n )ER +
Z
n) ] X
)
(3.8) goes over into the functional equation n
(3.13)
n
n
f.
i=1
=-
(t.) ~ ~
g (
~
t
t
.)G(~ ,
i=1 ~
tn
... , -
n-I
for the functions
fi,g:R+
density functions,
thus they cannot be almost everywhere
zero.
By (3.10)
R, G:R+
-
R.
the same applies to the functions
(i=I, ... ,n).
Using Lemma 3. I.,
it follows for all
that n
11
i=1
f.(t.)g(u l )G(u 2 , ••• ,u ~
~
-
219 -
n
)
¢
O.
are
fi,g,G
t.,U.ER+ ~
~
f.
~
Now, let
<
j
n
t.=t +t.
be fixed and
Then we get from (3.13)
if
J
n
~
i¢n,j.
that
n-I
f
n
(t
n
)f.(t.) J
J
n
i=1
f.(t +t.) ~
n
J
Hj
t.
g[(n-I)(t +t.)]G(I+-L J
n
for all
t
.,t ER n
J
tn
t.
-L t
, ... ,
t.
, ••• , 1 +
n
-L) t
n
. This implies the functional equation
+
t. J ) p(t .+t )q(-t
f.(t.)f (t )
(4 ' )
J
n
J
n
J
n
(t . , t ER ), J n +
n
where (3. 14)
g[(n-I)u] n-I f. (u)
p(u)
n
(uER+)
~
i=1 Hj
and (3.15)
q(v)
G(I+v, ... , v , ... ,I+v)
(vER+).
j '-'
Thus the functions
f.,f J
n
satisfy the condi-
,p,q
tions of Theorem 1.2 since measurable. Therefore
f.,f ,p,q
and
f.
J
n
are
are of the forms
J
b at j
A .t .e
J
J
(3. 16)
f where
n
(t
n
)
A .,A ERO J
n
1l
t
c-b-2 at n
n n
and
(t ER
e
n
a,b,cER
),
are arbitrary constants.
Then (3.9) and (3.10) gives
-
+
220 -
b A.!a'.(x.)![a.(x.)]
f . (x.) ] ]
]
]
]
]
]
e
aa.(x.) ] ]
(x.En ]
x.
),
]
(3.17)
aa f
n
A!a'(x)![a(x)]c-b-3+n e
(x )
n
n
n
n
n
n
(x
Furthermore
f.(x.) ~
~
The functions tions,
thus
=
0
x.ER,n
for
f.
~
(3.17) imply that
En
x
) n
). n
(i=I, ••• ,n).
x.
~
(i=I, ••• ,n)
~
n
(x n
are density func-
XI""'X n
ed gamma distributions with densities
have generaliz-
(3.6).
Thus the if part of our theorem is proved. (b) Let now XI""'X n be independent random variables with generalized gamma distributions (3.6). Using Lemma 1.3 and the Jacobian
(3.4) of the transformation
(3.2) we get for the density function mensional random variable ~
i= I
~
p.
i=1
p.-I
[F~I(YI)]i=1 ~
~
n
n
n-di-
n
!Fj[F -I I (Y I )]
n r (p . ) i =I ~ n-I
of the
the expression n
A
G
!
p .-1 [G. (Y 2 ' ••• , Y )] ~ ~
n
X --------~~~----------~n~----------------------n-I 1+ ~ G'(Y2""'Y)] i = I ~ n
It is easy to see that
G
-
~ (p.-I) i= I ~
!:H(y 2 ,
can be written as
22 I -
•••
,y n )!
where
n ~
A
n
p.
i= I
~
~
(p.-I)
~
[F-I(yl)]i=1
n
x
r(~p.)
i= I
~
(3.18)
and
for
(3. 19) n-I
p.-I
II [G'(Y2'''''Y )] ~
i=1
n II r (p,) i=1
~
n
n .~ Pi-I
n-I
~
[1+
~ G.(y 2 , .. ·,y] i=1 ~ n
(Y2, ... ,y)En X ... x n , i.e. n Y2 Yn • • •, y ) are independent • n Thus our theorem is proved.
for
~=I
1-H(y 2 , · · · , y )1, n
YI
and
(Y 2
, •••
Remarks
3.1. The formulas
(3.18) and (3.19) give the
density functions of random variable • • • , Y)
n
YI
in the forms for
(3.20) otherwise and - 222 -
and
(Y 2 , •••
(y2 •... ,y)En x .•. xn n Y2 Yn
for (3.21 )
otherwise respectively. (3.18) and
(3.20)
shows that
has also a -1
generalized gamma distribution (1) a(yI)=F I n and parameters A and ~ p .• i= 1 ~ Further if a.(x.)=x. (i=I, •.. ,n), ~
~
(Y I )
we
~
get that the sum of gamma distributed random variables is gamma distributed, which is well-knwon (see e.g. 3.2. Let all
xER+.
and
n=2
F.(x)=x, ~
Then Theorem 3.1
a.(x)=x ~
n=:<
and
for
gives the well-knwon cha-
racterization of gamma distributions 3.3. Let
(i=I,2)
[6J).
FI (x)=x,
F2
(see e.g.
[I5J).
is arbitrary
satisfying the conditions of Theorem 1.3. Then Theorem 3.1
is a generalization of a theorem of I. KOTLARSKI
(see [9J). 3.4.
If
a(xj=x a
(xER+,
>
a
0), then (1)
is the
generalized gamma distribution introduced by AMOROSO [2J
(see also [8J,
[I3J,
[I7J,
[I8J,
[I9J).
REFERENCES [IJ
J. Aitchison,
Inverse distributions and independent
gamma-distributed products of random variables, Biometrika, 50(1963), 505-508.
[2J
L. Amoroso,
Ricerche
interno alIa curva dei redditi, 2(1925),
Annali dj Mathematica,
[3J
J.A.
123-159.
Baker, On the functional equation
f(x)g(y)
154-162.
n II h.(a.x+b.y),
i=I
~
~
~
-
223 -
Aeg. Math.,
11(1974),
[4J
J.A.
Baker, On the functional equation
= p(x+y)q(~), y
[5J
Aeq. Math.,
f(x)g(y)
14(1976), 493-506.
P. Flusser, A generalization of a theorem by M.V. Tamhankar, Journal of Multivariate Analysis,
1(1971),
288-293. [6J
N.C. Giri, Introduction to probability and statistics, Marcel Dekker, New York,
[7J
E. Hewitt - K.A. Ross, Vol.
[8J
1974.
Abstract harmonic analysis,
I, Academic Press, New York,
1963.
H. Jakuszenkov, On properties of the generalized gamma distribution, Demonstratio Math., 7(1974), 13-22.
[9J
1.1. Kotlarski, Una caratterizzazione della distri-
buzione gamma per mezzo di statistiche indipendenti, Rendiconti di Matematica,
2(1969), 671-675.
[10J K. Krolikowska, On the characterization of some families of distributions, Comment. Math.
Prace Mat.,
17(1973), 243-261. [IIJ K. Lajk6, Remark to a paper of J.A.
Baker, Aeq. Math.
(to appear). [12J K. Lajk6, On the functional equation =h(ax+by)k(cx+dy), Periodica Math.
f(x)g(y)=
Hungar.,
(to
appear) .
[13J J.H. Lienhard - P.L. Meyer, A physical basis for the generalized gamma distribution, Quart. Appl. Math., 25(1967), 330-334.
[14J E. LukAcs, A characterization of the normal distribution, Ann. Math.
Statist.,
- 224 -
13(1942), 91-93.
[15J E. LukAcs, A characterization of the gamma distribu-
tion, Ann. Math. [ 1 6 J I.
0 1 kin,
[17J A.C.
Statist.,
P 128, Aeq. Ma th.,
12 ( 1 975),
Statist.,
290- 292.
9(1938),
176-200.
Roslonek, On some characterization of the gen-
eralized gamma distribution, Warsz.,
[19J E.W.
tion, [20J M.V.
173(1968),
Zesz.
Nauk.
Politechn.
127-134.
Stacy, A generalization of the gamma distribuAnn. Math.
Statist.,
33(1962),
1187-1192.
Tamhankar, A characterization of normality,
Ann. Math.
K.
319-324.
Olshen, Transformations of the Pearson type III
Distribution, Ann. Math. [18J E.
26(1955),
Statist.,
38(1967),1924-1927.
Lajk6
Mathematical Institute of the L. 4010 Debrecen, Pf.
12., Hungary
Kossuth University
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION HETHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
ON SOME PROPERTIES OF SYMMETRIC STABLE DISTRIBUTIONS* E. LUKACS
INTRODUCTION D.
DUGUE proved the following theorem (see [3J):
THEOREM A. Suppose that the random variables
X
and
Yare normally distributed with zero mean and unit variance and that
! = (Zl,Z2)
is a bivariate random vector
with non-negative components and distribution F(zl,z2).
Assume that
X,Y
and
Z
function
are independent and
let
w
X~-yrz-;IZ 1 +Z 2
Then
W
is normally distributed with zero mean and unit
variance for arbitrary
F(zl,z2).
In this paper we discuss an extension of Theorem A ...':
The preparation of this paper was supported by the Fonds zur Forderung wissenschaftlicher Forschung under a project directed by G. TINTNER.
- 227 -
and also investigate its stability. In Section I Theorem A is generalized, in Section 2 we make first some remarks concerning the closeness of distributions and study then the stability of the theorem.
In Section 3 we generalize
Theorem I and obtain a characterization of symmetric stable distributions. I.
GENERALIZATION OF THEOREM A
THEOREM I. Let X and Y be two random variables which have both the same symmetric stable distribution with exponent
(0 < a
a
Suppose that
Z
~
2)
and scale factor
A > O.
is a bivariate random vector
(ZI,Z2)
with non-negative components and distribution
F(zl'Z2)' Assume further that
X,Y
and
Z
function
are indepen-
dent and let ZI Ita Z2 Ita (Z +Z) X-(z +Z) Y.
w
I
2
I
Then the random variable
2
has also a symmetric stable
W
distribution with exponent
a
and scale parameter
A
F(zl,z2)'
for arbitrary
PROOF. We denote the distribution function of the symmetric stable distribution with exponent parameter
A
a
and scale
SaA(x) so that the corresponding
by
characteristic function is co
saA (t)
= J -co
Let
v(u)
e itx dS aA
(x)
be the characteristic function of
v(u)
=
E(exp(iuw»
-
228 -
W.
Then
J J
(I. I)
o 00
0
z
00
I/a a
z2 I/a a \ -A \-u(--) \ 1dF(z I ,z2)· zl+z2
J J exp[ -A \ u(_ _ 1 -)
o0
zl+z2
are non-negative this becomes
and
Since
v(u)
hence
v(u)
so that Theorem
= exp(-A\u\a)
is proved.
Theorem A is obtained as a particular case by putting
I
a=2, 1..=2. REMARK I. W
is independent of
Z.
REMARK 2. The statement of the theorem remains true if
W
is replaced by
Z lIla [Z lIla [ZI+!2 x+ ZI!Z2 Y. 2. THE STABILITY OF THEOREM It is very often convenient to introduce a metric in the space of distribution functions. This can be done in a variety of ways.
In this paper we shall use the un-
iform (Kolmogorov) metric where the distance between two distribution function p(H,F)
sup
Hand
is defined as
F
\ H(x) -F (x)
!.
x
DEFINITION I. Let
Hand
F
functions, we say that
Hand
Fare
other if
p(H,F)
~
Ce
where -
C
229 -
> 0
be two distribution e-close to each
is a constant inde-
pendent of
£.
DEFINITION 2. Let
Hand
functions. We say that tion F
is an
£-contaminated distribu-
with a contaminating function (I-£)F+£K
(i)
H
(ii)
K
interval
<
£
I)
x.
be a function of bounded variation. We
(V~ G)
VbG a
<
(0
if
K
is a function of bounded variation in
G(x)
Let write
H
be two distribution
F
G
for the total variation of
-~
in the
([-~,~]).
[a,b]
THEOREM 2. Suppose that the distribution function is an
H
£-contaminated distribution function
contaminating function
K, then
Hand
F
Fare
with £-close
to each other.
PROOF. It follows from the assumption that £K-£F, hence
and
Fare
where
!H-F! S £(C+I)
C
=
H-F =
V~ K, -~
i.e.
H
£-close to each other.
THEOREM 3. Suppose that the common distribution function
an
of the random variables
H(x)
X
and
Y
SaA(x)
£-contaminated symmetric stable distribution
with contaminating function
K(x).
Let
~
=
is
(ZI,Z2) be a
bivariate random vector with non-negative components and distribution function Z
F(zl,z2)' Suppose that
X,Y
and
are independent. Let W
=
ZI
I/a
Z2
I/a
x(z +Z)
-Y(z +Z )
I
I
2
2
be a random variable with distribution function then
-
230 -
V(x),
O(e: log
..!..) e:
(as
e: -
0)
PROOF. For the sake of simplicity we write for
Sa)..
sa)..(t)
(x)
of
Sex)
set)
for the characteristic function 00 itx dK(x) for Sex). We also write k(t) = J e and
-00
the Fourier-Stieltjes transform of the contaminating
K(x).
function
K(x)
Since
is, by assumption, a function of boun-
ded variation with total variation
Ik (t) I
(2. J)
s c
=
V-00 K. OO
The characteristic function
vex)
of
v(u)
of the distribution
w is then v(u)
(2.2)
c, we see that
00
E[ exp(iuw»)
= 00
00
I I{I o 0
-00
=
Q)
I
e
iuw
dH(X)dH(y)}dF(zl,z2)
-00
where (2.3)
Since
H(x)
(1-e:)S(x)+e:K(x)
we have
v(u)
(2.4)
I IU
o
0
-00
I
e iuw [ (1-e:)dS(x)+e:dK(x»)X
-00
X[ (I - e: ) d S (y ) + e: d K ( Y )] } d F ( z 1 ' z 2) •
-
231
-
5ince
(2.5a)
f eiuxds(x) it follows that
s(u)
f exp[iux( Z
-00
I
ZI
ZI I/o. + ) ]dS(x) Z
s[
2
+
U ( Z
I z2
I/o. )
]
similarly
z2
00
(2.5b)
f exp[iuy(
zl+ z 2
Z2
I/o. ) ] dS(y)
s[ -u ( Z
I
+
z2
I/o. )
Therefore, on account of (2.3)
f
f exp(iuw)dS(x)dS(y) =
-00
or, since
zl
and
are non-negative,
00
(2.6a)
f
f exp(iuw)dS(x)dS(y) = exp(-Alulo.),
in the same way we obtain the relations 00
f (2.6b)
f exp(iuw)dS(x)dK(y)
=
ZI I/o. z2 I/o. s[u(z +Z) ]k[-u(z +z) ] I
2
I
2
00
f
f exp(iuw)dK(x)dS(y)
-00
-00
(2.6c)
- 232 -
z
]
00
J
J exp(iuw)dK(x)dK(y)
(2.6d)
It follows from (2.4),
(2.6a),
(2.6b), (2.6c) and (2.6d)
that
00 00 zIlla z2 Ila +£(I-£)J J s[u(---+---) ]k[-u(---+---) ]dF(zl,z2)+ o0 zl z2 zl z2 00 00 zIlla z2 Ila + (I-£)£1 J k[ u(---+---) ] s[ -u(---+-) ] dF(zl ,z2)+ o0 zl z2 zl z2
+
£
200 00 zIlla z2 Ila J J k[ u ( - - ) ] k[ -u(-+-) ] dF(z I ,z2)· o 0 zl+ z 2 zl z2
In view of (2.1) and the fact that all characteristic functions have modulus not exceeding unity we get 2 2 2 jv(u)-s(u)j :;; 2£+£ +2£(I-£)C+£ C or jv(u)-saA(u)
(2.7)
j
:;;
2 2 2(I+C)£+(C-I) £ •
We have now to apply a result which is due to L.D. MESHALKIN and B.A. ROGOZIN [2J. We formulate it as the following lemma. LEMMA. Let let
G(x)
be a
F(x)
that for arbitrary and
G
be a nondecreasing function and
function of bounded variation. A,T
and
£
>
0
the functions
satisfy the following conditions:
-
233 -
Suppose F
(i)
F(-OO) = G(-OO) ,
(ii)
G' (x)
exists for all
x
and
1 G'
(x)
1
s
A,
00
where
J
f(t)
eitxdF(X),
-00 00
J
get)
eitxdG(x).
_00
(iii)
If(t)-g(t)
1
<
for
E
It follows that for an arbitrary
(2.8)
IF(x)-G(X)
<
1
is satisfied where y (L)
VOO
VbG(x)
and
(2.9) Here
a
function
-00
C
1t 1
<
> 21T
L
C[Elog(LT)+ ;
the inequality
+y(L)]
is an absolute constant while
G(x)-supV
x+L G(y). x
VOO G(y)
denote the variation of the
x
-00
G(x), over the interval
(a,b)
respecti vely.
In order to apply the lemma we put (2. lOa)
F(x)
=
V(x),
(2. lOb)
G(x)
=
Sex).
After a simple computation we see that IS'(x)1
T.
s
so that (2.IOc) A
We have
-
234 -
and
(-00,+00)
L/2
L
-L
S(-)-S(-) 2 2
V_ L / 2 S(y) Therefore x+L'
supV x
L/2
~
S(y)
V_ L / 2 S(Y)
x
=
L
2S(2)-I.
We see then from (2.9) and (2.IOb) that (2.11)
x+L
y(L)
We select (2.12a) 0
E
l-supV x
L/2
S(y) S I-V_ L / 2 S(y)
so small that
< EI < min{2CI-S(I)J,
and choose then
A,
I}
so that
L
Then by (2.12a)
S(I) <
(2. 13)
E
1-
I
T
and it follows from (2.11) that (2.14) We see from (2.12b) and (2.13) that LEI
S(2) = hence (2.12c)
L
1-
: t > S(I)
> 2. Finally we select T = A/E
I
> I.
This is possible for sufficiently small
- 235 -
E. Then
(2.12d) LT
>
2
as required by the conditions of the theorem. Since
L
is finite,
there exists an integer such
that'-'
(2. 15)
L
< (_I ) E:
n
I
The conditions of the lemma are satisfied and we see from (2.8)
that
I vex) -Sat.. (x) I It follows
2
~
c[
from (2.12c)
<
LT
<
AE:
E:
Ilog (LT)
and
(2.15)
+~Y (L)] that
-(n+l) I
so that log 2
In view of
<
log
(2.IOc),
(LT)
<
log A+(n+l)log E:
(2.12c)
and
+ E: log [ I
I
-t..
I
(2.14) we have
- I /a
I r (-+ I )] + 2 E: I}
a
It
so that
*
In case
x=2, that is in the case of the normal distribution
one can use the inequality
I-~(x) < x-1cp(x)
[IJ, pp. 175) to determine
n
(cp
(x
> 0)
is the standard normal
density). In this case a simple computation yields
-
236 -
(see
L
< 2V~. ltE:
1 :S C{(n+I)£l log - + £1
P(V,saA)
1 - 1/ a + £ 1 log [ itA
1 (a-+ 1 )] + 2 £ 1 }
r
that is _I) O(£l log £1
P(V,SaA)
Noting that
(£1- 0).
is given by (2.7) we obtain the state-
£1
ment of Theorem 3.
3. GENERALIZATION OF THEOREM 1 AND CHARACTERIZATION OF SYMMETRIC STABLE LAWS THEOREM 4.
Let
X 1 'X 2 " " ' X n
be
n
identically
distributed random variables having a symmetric stable distribution
SaA(x).
Suppose that
Z
is a random vect02 whose components are either all nonnegative or all ncn-positive and denote the distribution function of
Z
X 1 'X 2 " " ' X n
and
j=1
variable itrary
Z
n Z, ~ (/
W =
where the
F(zl,z2""
by
6, ]
W
ZI
'Zn)'
Assume further that
are independent and let I/a )
X ,6 ,
.. 'Zn
]
are either
]
or
+1
-I.
has also the distribution
Then the random for arb-
F(zl,z2"",zn)'
PROOF. We assume first that all the components of
z
are non-negative and write
for all Z
j=I,2, ... ,n}.
R
+
n
=
{(zl""'z n ); z,]
~
The case where all component of
are non-positive is treated in the same way. -
237 -
0
v(u)
Let again
the random variable
be the characteristic function of W.
Then
00
v(u)
= J+U ... J R
n
n z. I/o. exp[iu ~ ( ] ) x.o .Jx . I Z 1+ ••• +z ] ] ]= n
Therefore n J+exp(-Alulo. ~
v(u)
R
Z.
+ ] + )dF(zl'···'z) n j=1 zl ••• zn
n
THEOREM 5. Let
X I 'X 2 ' ... 'X n be n independently and identically distributed random variables with distr~ bution function
Z
Let
=
G(x)
and characteristic function
(ZI'Z2 .. . Zn)
nents have all the same sign and let
are independent.
, Z
n -
n ~ (
W =
j=1
where
0
<
0.
further that
<
z. zl+···
W
x I ,X 2 ,
...
Let
I/o.
]
and
2
F(zI ,z2'··· ,zn) be
Z. Suppose that
the distribution function of •• • ,X
get).
be a random vector whose compo-
+
zn
)
0 .X . ] ]
o.]
is either
+1
has the same distribution
X.'s for any distribution
F,
]
then
G(x)
or
-I.
G(x)
Assume as the
is symmetric
stable distribution with exponent
PROOF. We assume that all Zj S 0
for all
j
Z. ]
~
0, the case where
is treated in the same way. -
238 -
+ = { ( z l " " ' z n ):Z.] ~ 0 for j=I,2, ••• n .. . ,n}. According to the assumptions of Theorem 5 W
We write
R
g(u)
has the characteristic function
=
E(exp(iuX I ))
that is
g(u) n
J+ J ... J exp[iu ~ ( j=1
R
n
where
dGdF
=
or
dG(xl) ••• dG(xn)dF(zl"",zn)
n (3. I )
z. I/a. + ] + ) x.o.}dGdF zl ••• zn ] ]
g(u)
.
z.
n
J=
I
glue
zl+"
I/a.
J) .+z
o.}dF(zl ••••• z ) . ] n
n
o
~n
Let ~I ~ O. ~2 ~ O. ~3 numbers and select
be
n
real
(3.2) where
£(z)
is the degenerate distribution which has a
single saltus at
Z
=
O. We substitute (3.2) into (3.1)
and get the functional equation g(u)
We note that teristic function) see that (3.3)
g(u)
=
g(O) = I
(since
and put
~I
> 0,
g(u) ~2
=
is a charac0,
°1
=
-I
and
g(-u).
It is therefore no restriction to assume that ° 1=° 2 =1 and we see that the characteristic function is real and satisfies the functional equation
- 239 -
g(u)
(3.4)
g[
~
(u CX
~1 l/cx + ~) 1 g[ 1 2
~2
(u CX ~ + ~) 1
l/cx
1
2
We introduce the function
h(u)
(3.5)
and
g(u 1 / CX )
=
(3.4) becomes
h(u cx )
~1
= h(u cx
We substitute here
~
u
1
+~
)h(u
CX
~2
)
~. 1 2
2
= (~1+~2)1/CX
and get
This equation has the solution
h(u)
e
where
is a constant.
g(u) S 1
Since
>
eu It follows then from (3.5)
g(u)
(3.6)
(A
e
0)
e
we see that
is negative, e
so that
g(u)
=
e
_AU CX
>
(u
0)
and we conclude from (3.3) that
g(u) which is the statement of Theorem 5.
-
240 -
-A
REFERENCES [1]
W. Feller, An introduction to probability theory and its applications, Vol. 1 (Third edition), J. Wiley &
Sons, New York, [2]
1968.
L.D. Meshalkin - B.A. Rogozin, Estimation for the distance of distribution functions based on the closeness of their characteristic functions and its application to the central limit theorem (in Fussian), I z d at. Ak ad. Nauk Uz b e k.
[3]
S S R , T ask e nt, 1 963, 49 - 5 5 •
D. Dugue, Variables scalaires attachees
a
deux
matrices de Wilks, Comparaison de deux matrices de wilks en analyse des donnees, C.R. Acad. Paris, 284(1977), 899-901. E. Lukacs 3727 Van Ness Str. NW Washington, DC 2016, USA
-
241 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION llETHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
A CHARACTERIZATION OF A GENERAL CLASS OF MULTIVARIATE DISCRETE DISTRIBUTIONS J.
PANARETOS
1.
INTRODUCTION
Let
X,Y
variables
be two non-negative integer-valued random
(r.v. 's). RAO and RUBIN [IJ have shown that if
ylx
the conditional distribution of parameters
nand
P(y
( 1. 1 )
p, n)
is binomial with
i.e. r n-r ( n) r p q (r=O,I, ... ,n; n=O,I, ... )
with
p
a fixed number lying in
(0,1), q=I-p, then the
Rao-Rubin condition (R-R condition), namely, ( 1 .2)
P(y
r)
P(y
r
Ix
y)
(r=O,I, ... )
holds if and only if (iff) the distribution of Poisson. Later, other authors SRIVASTAVA and A.B.L.
X
is
(e.g. TALWALKER [4J, R.C.
SRIVASTAVA [3J) used the
R-R
condition to obtain characterizations for other discrete distributions. They have also extended some of the results -
243 -
to the multivariate case. SHANBHAG [2J
gave a generalization of Rao and
Rubin's result using a technique existing in the renewal theory.
Shanbhag's result provides most of the relevant
results existing in the literature as special cases. In Section 2 of this paper we state the result of Shanbhag. Our main result, i.e. the multivariate extension of Shanbhag's result, is presented in Section 3. Finally, in Section 4 we illustrate our method by obtaining characterizations of some well-known multivariate discrete distributions. We also point out that an improved version of Ta1wa1ker's characterization [4J of the multiple Poisson distribution is a corollary of our main result. 2. SHANBHAG'S EXTENSION OF THE R-R CHARACTERIZATION LEMMA I
(Shanbhag [2J). Let {(V ,W ), n=O, I, ... } n n be a sequence of vectors with non-negative real components such that
Vn~O
for some
n
~
I,
WI~O.
Then
00
(2. 1 )
V
m
n=O
i f f for some V
V
~
b
>
(m=O,I, •.• )
W n+m n
° (n= I ,2, ••. ),
n
(2.2) 00
~
n=O
wn b n
=1.
As a result of Lemma I, Shanbhag obtained the following theorem.
-
244 -
be a sequence ,b ):n=O,I, ... } n and an > for every n 2: {c } the for n 2: 2. Denote by
{(a
THEOREM I. Let of real vectors with bO
>
0,
>
b l
0,
°
°
b n 2: {a }
°
n
{b}. n n be a random vector of non-negative in-
convolution of Let
n
(X,Y)
and
teger-valued components such that with
and whenever
(2.3)
rlx
P(Y
P
C
n
>
n
°
a b r n-r
n)
C
Then the R-R condition
(2.4)
P
P(x=n)
°
Pn , n 2:
=
we have
(r=O,I, .•. ,n).
n
(1.2) holds i f f for some
8
>
°
(n=I,2, ... ).
n
PROOF. This follows from Lemma I if one defines the sequences
V
n
, W
n
P
(2.5)
V
n
C
by
n
a
(n 2: 0) ,
b
W
n
n
n :E P n n C n=O n
These sequences satisfy all conditions set by Lemma I. On the other hand it can be checked that alent to
(2.1) and (2.4) to
(1.2) is equiv-
(2.2).
REMARK I. Theorem I provides characterizations for many well-known discrete distributions such as
the
Poisson, binomial and negative binomial.
3. THE MULTIVARIATE EXTENSION THEOREM 2. Let
{(a,b): n n n =0,1, ... ; i=I,2, ... ,s; s;;-I,2, ..• } vectors such that
i=I,2, ... ,s
with
a
n b-;;
> 0, > 0,
-
b
n
2:
°
b-
=
(nl,···,n s )'
-
i
be a sequence of real
for every
0, ..• ,0,1 >
245
n =
°
and
n. 2: 0, ~
some
b 00, . . . "
••• ,
some
b
° I ,n s
>
>
l,n 2 ,n 3 ,···,n s
convolution of
{a} n
b
some
0,
0.
gi ven by
C
n
a
r
~
and
rl,···,rs
=
n
denoting
r=O
0,
n
n
a
where
>
to be the
{C } n
Define
{b } n
and
0,0, . . . , l,n s _ 1 ,n s
~
a b
r=O r n-r
n I
n2
~
~
rl=O r2=0
5
~
r
5
=0 Consider a
! =
" " X s )'
random vector
(YI'''''Y s )
~
where
with
Xi'
Yi
for every
>
l , · · · ,n s
i=I,2, . . . ,s
°
and whenever
(X I ' ••
(i=I,2, . . . ,s)
non-negative integer-valued r.v.'s such that
= n l , •. . ,x = n ) = P s s n
=
for some P
n
>
Pn
n.
~
P(X I
=
and
°
a b
r n-r
(3. I)
P(Y
!:.)
C
n
(r.=O,I, ... ,n.; i=I,2, . . . ,s) ~
Also define
( .) X]
(j=2,3, ••• ,s)
=
(XI, . . .
and let
k=I,2, •.• ,j-1 (3.2)
P (!
i f f for some P
(3.3)
Also i f
C
r)
Po
n
Co
( 3.3)
X. ]
y(j) ]
E..I~ = !)
P (r
E..lx(j)
>
s IT i-I
denote that] (X k
> y.). Then
p (!
8 1 ,···,8 s
n
,x.), y U ) = (yl, . . . ,Y.)
x(j); and
~
>
y(j) )
(j=2,3, . . . ,s)
°
n.
e.
~
~
is true then
-
Y
and
246 -
X-Y
are independent.
PROOF. If we use the notation
P(y
=
£lx(O) = y(O)) =
P(r
x(O)=y(O)
to denote
we can see that
£)
(3.2) is
equivalent to
P(r
(3.4)
= £I~ =
P(r
=
r)
=
Elx U - 1 )
=
(Q,=1,2, ••• ,s).
Now define the sequences
V
n
s
(3.5)
b
0,0, ••. ,O,n s
P (x -
=
!:)
W
n
for fixed
s
>
r.
~
0, i=I,2, ... ,s-1
case we have that for V =V W I: n +r r n n =0 s s s s s
Q,=s
and
s
Q,=s
p
r1,···,rs_1,n s
r1,···,rs_1'0 C
r1,···,rs_1,n s
r1,···,rs_1'0
for some
r.~
n
>
s
0
O.
In this
and hence using Lemma 1 we come to
p
every
~
(3.4) is equivalent to
the conclusion that (3.4) holds for
C
n
and every
r.
~
>
0
iff n
o
s
0s
s
>
0,
(i-l,2, . . . ,s-l)
(since
were fixed but arbitrary). Consequently (3.4) for
Q,=s
holds iff p
(3.6)
C
n
n
p
n1, ••• ,ns_1'O C n1,···,ns_1'O
-
n
os s
247 -
for some
0
>
0
and
°
(i=1,2, .•• ,s). It can also be verified every n.~ > that whenever (3.6) is valid we have that, conditional on x(s-I) = y(s-I), y and x -y are independent. s s Let us now define the sequences p
(3.7)
C
r l , . . . ,r£_1 ,0, . . . ,0
(r.
~
r l , •.• ,r£_1 ,0, . . . ,0
i
and every
n£
>
fixed,
0
= 1,2, . . . ,£-1)
and
0
l:
(3.8) n
n X
'"
8£+1'"
holds for
s
=0 p(~
n £+ 1
"'£+1
for
>
···'0 s
>
.,8 s £=k,
'" s
b
o
0,
= !.)
p(x(£-I)
Assume that
£=1, . . . ,s-I.
k+I, ... ,S; 2
~
k
~
(3.4)
and is equivalent
s
to p
p
n l ,·· .,nk_l,n k , · · · ,n s
(3.9)
C
C
n l ,·· .,nk_l,n k , · · · ,n s
n I ' ••• , n k _ I ,0, .•. , 0
n X
for some
Elk""
(Note that if =y(k-I)
,8 s
(3.9)
>
0
and every
is valid then,
x
n I ' ... , n k _ I ,0, .•. ,0 8
k
n.
~
k •••
>
0
n El s
s
(i=1,2, . . . ,s).
conditional on
jk-I)=
and (Xk-Y k , Xk+I-Yk+I""'Xs-Ys) are independent.) Under these circumstances it can be shown
that,
,
y
for
£=k-I,
(3.4) is equivalent to
p
(3.10)
p
nl,···,nk_I,···,n s C
n l ,···,nk _ 2 ,O, •.. ,0 n k _ 1
C
nl,···,nk_I,···,n s
-
n l ,···,nk _ 2 ,0, ••• ,0
248 -
Elk-I
x
for some
8 k _ I ,···,8 s
... , s; 2
~
k
~
and for every
n.
~
>
0
(i=1~2,
•••
This is so because with the help of
s) •
Lemma I we can see that, for
(3.4) holds iff
R.=k-I,
p
nl,···,nk_I,O, ... ,O
(3. II)
C
nl,···,nk_I,O, ... ,O ~
(2
i.e.
k
~
s)
(by combining (3.9) and (3.11», iff (3.10) holds.
We may also observe that if (3.10) is valid then conditional on x(k-2)=y(k-2), K and (Xk_I-Y k _ 1 ,Xk-Yk , · · · .. . ,x -y ) will be independent (2 ~ k ~ s). s
s
Consequently, we can say that (3.2» (i. e.
(3.4)
(and hence
is equivalent to (3.3). Also we have that if (3.3) (3. 10) for
k=2)
holds, Y
and
X-Yare indepen-
dent. Hence Theorem 3 is established. 4. CHARACTER:ZATION OF THE MULTIPLE POISSON, BINOMIAL AND NEGATIVE BINOMIAL DISTRIBUTIONS As a result of Theorem 2 the following corollaries can be established. COROLLARY I
(Characterization of the multiple (~,!)
Poisson). Suppose that for the random vector know that s
(4. I )
P (K
~)
n i=1
n.
(~)
r. n. -r. ~
r. Pi qi
~
~
~
n.
~
i=1,2, ••• ,s)
- 249 -
~
0,
we
(i. e. multiple binomial) then condition (3.2) holds i f f n. ~ s s A. -A ~ e (i=I, ... ,s; A= ~ A.,A. > 0) P n (4.2) n ~ i= 1 ~ ~ i=1 ~ (i.e.
multiple Poisson).
PROOF. Observe that (4.1) is of the form (3. I) with n n.~ s s P. i qi ~ and b a n ~ n ~ (n ~. =0, 1 , ••• ) • n n ~ ~ i=1 i=1 s
Since the corresponding
C
n
n i=1
:n:-r
for
n.
~
~
0
the
~
Corollary follows. REMARK 2. TALWALKER [4J derived a similar characterization of the mUltiple Poisson distribution using a condition similar but more complicated than our condition (3.2). COROLLARY 2
(Characterization of the multiple
binomial). Supppose that form
P
is mUltiple Poisson of the n (4.2) and that the conditional distribution of ~1~
can be written in the form is true i f f
P(~ = £I~ =~)
(3.1).
Then condition (3.2)
is multiple binomial of the
form (4.1).
PROOF. The necessary part of the proof is straightforward and is contained in Corollary I. For "sufficiency" we observe that Theorem 2 implies that condition s n. (3.2) holds iff c = c n (A.e.) ~/n.!.Using TEICHER's n 0 i=1 ~ ~ ~ [5J extension of Raikov's theorem we see that this is so s s n. n. iff a a O n (a..) ~ In.! and b b O n (8.) ~ In.! n ~ n ~ i=1 ~ i=1 ~ {a }, {b } Since should (a. i ' 8.~ > O·, a..+8.=A.e.). ~ ~ ~ ~ n n -
250 -
satisfy the latter conditions it is immediate that we !I~
should have the distribution of
to be multiple
binomial of the form (4.1), for some COROLLARY 3
(PI, •.. ,ps)E(O,I).
(Characterization of the multiple
negative binomial). Suppose that the vector
is
such that
p (!
(4.3)
!2..)
i=1
(
-m -p i i) n. ~
(r.
$
~
n.; m.,p. ~
~
~
(i.e.
multiple negative hypergeometric).
(3.2)
holds i f f
p
n
>
0,
i=I,2, •.• ,s)
Then,
condition
is mUltiple negative binomial of
the form
(4.4)
p
(N.=m.+p.).
n
~
~
~
PROOF. The proof follows easily if one observes that (4.3) is of the form (3. I) with s
a
n
m.+n.-I
n
(~
~
ni
i=1
n.
)q.~ ~
(4.5)
s
b
n
p.+n.-I
n
(~
~
ni
i=1
s
in which case
C
n
n.
)q.~ ~
m.+p.+n.-I
n(~
i=1
~
~)
251
~
qi
ni
-
n.
-
.
REMARK 3.
It is clear that for different forms of
the sequence {a,b} characterizations for other forms n n of multivariate distributions can be obtained. Acknowledgement.
I am grateful to Dr. D.ll. SHANBHAG
for his valuable comments and helpful discussion on the subject. REFERENCES [IJ
C.R. Rao - H.
Rubin, On a characterization of the
Poisson distribution, [2J
D.N.
Sankhya A,
26(1964), 295-298.
Shanbhag, An extension of the Rao-Rubin cha-
racterization of the Poisson distribution, J. Prob.,
[3J
R.C.
Appl.
14(1977), 640-646.
Srivastava -
A.B.L.
Srivastava, On a cha-
racterization of Poisson distributions, J.
App.
Prob.,7(1970),497-501.
[4J
S. Talwalker, A characterization of the double Poisson distribution, Sankhya A, 32(1970), 265-270.
[5J
H. Teicher, On the multivariate Poisson distribution, Skand.
Aktuartidskr.,
J. Panaretos 8 Cratesicleias St. Athens 504, Greece
-
252 -
37(1954),
1-9.
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION UETHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
A CHARACTERISTIC PROPERTY OF CERTAIN DISCRETE DISTRIBUTIONS J. PANARETOS -
E. XEKALAKI
INTRODUCTION
I.
A shifted univariate distribution has a probability
skG(s)
generating function (p.g.f.) of the form
G(s)
is the p.g.f. of a distribution on the integers
G(s)
0,1,2, . . . . The distribution with p.g.f. to be shifted as
where
k
is said
units to the right or left according
is a positive or negative integer.
k
In the bivariate case a shifted distribution will have p.g.f. of the form
k
m
s t G(s,t), where
G(s,t)
represents the p.g.f. of a distribution on X
{O,I, ••• }
and
k,m
{O, I , ••• } X
are integers.
Consider now two distrete random variables
x
Y. Assume that
and
Gy(s)
~
k
s Gx(s),
k
(r.v.
integer.
Then it can be shown that the factorial moments of relate to the factorial moments of r
(1. I)
L
(l)k(i)E(x(r-i))
i~O
where
z
(r)
X
z(z-I) ... (z-r+I), -
253 -
y
thus (r=0,1,2, ... )
IS)
Analogous is the expression
for the factorial mom-
ents of the vectors X = (X I 'X 2 ) and Y = (Y I ,Y 2 ) k m s t Gx(s,t) G (s,t) (k,m integers), i. e.
with
Y
( I .2)
(r=O,I,2, ••• ,
J1.=O,I,2, ••• ).
For certain types of discrete distributions the relationships between the factorial moments of their original and shifted forms reduce to expressions which can be shown to constitute a unique property. In the sequel, such properties will be used to provide characterizations for some well-known discrete univariate and bivariate distributions. Specifically, in Section 2 we provide a characterization for the geometric which subsequently is extended to characterize the class of distributions which consists of the Poisson, binomial and negative binomial distributions. A characterization of the Hermite distribution is also given. Section 3 extends the results to obtain characterizations for some bivariate distributions whose marginals are independent. Finally, Section 4 considers the case of certain bivariate dependent distributions. 2. CHARACTERIZATION OF SOME UNIVARIATE DISCRETE DISTRIBUTIONS THEOREM 2.1. Let r. v. ' s
X,Y
be non-negative discrete
such that
(2. I)
where tion
Gw(s)
denotes the p.g.f. of w. -
254 -
Then the condi-
tion
(2.2)
>
(c
is necessary and sufficient for parameter
PROOF.
c
X
r= 1 ,2 , ... )
I,
to be geometric with
-I
Necessity follows immediately.
From (1.1) we have for
Sufficiency.
k=1
(r=I,2, .•. ).
(2.3) Hence (2.2) holds if and only if (iff)
(r=I,2, ... ), i.e.
iff
o
(r=O,I,2, ... ),
which implies that (r=O, 1 ,2 , •.. ) . But this is the
r-th factorial moment about the origin
of the geometric distribution with parameter
q=c
-I
Hence the theorem is established. Note.
In the context of stochastic processes, the
characteristic property (2.2) is equivalent to the well-known lack-of-memory property (see PARZEN [3J, p.123). It has just been proved that the geometric distribution is uniquely determined by E( (x+ I )(r))
cE(x(r))
-
(r=I,2, ••.
255 -
c
>
I).
One may ask what other distributions can be characterized by similar properties. Consider for example the more general case where
c
is not a constant but
instead it is a function of
r. Specifically, let
X
be
a non-negative discrete r.v. with the property that (r=m, m+ I , ... ) for some positive integer m=1
and
C(r)=ar+b, a,b
m. Consider the simple case
>
0, i.e. (r=I,2, ... ).
(2.4)
What distributions can be characterized by this property? By the following theorem it turns out that
(2.4)
uniquely determines the class of distributions which contains precisely the Poisson, binomial and negative binomial distributions. THEOREM 2.2 (univariate case). Let Theorem 2.1.
holds i f f
X
=
(r=1 ,2, •.• ; a,b
(ar+b)E(X(r-I))
>
0)
has one of the following distributions
(i)
Poisson with parameter
( i i)
binomial with parameters for
(iii)
be as in
Then the condition
E(y(r))
(2.5)
X,Y
a
<
b
for
a=l.
p=l-a, n=-I+
I,
negative binomial with parameters and
b
I-a
b
k=l+ a-I
for
a
>
q=(a-I)/a
I.
PROOF. Necessity follows immediately. Sufficiency.
From (2.3) we have that
iff - 256 -
(2.5) holds
= 0
E(x(r»-[(a-l)r+b]E(X(r-I»
(r=I,2, •.. ),
Le. iff (2.6)
E (x (r+ I) ) _ [ (a-I) r+a+b-I] E (x (r) )
Case
o
(r=O,I,2 ..• )
a=l. Then (2.6) becomes
o
(r=O, I ,2, ••• ) •
Solving we obtain E(x(r»
= br
which implies that Case
a~l.
(r=O,I,2 •... ) X
~
Poisson (b).
We have from (2.6)
(r=O, I ,2, •.• ) .
Solving we find that (r=O, I ,2 , ••• )
(2.7)
where
z(r) = z(z+I) ... (z+r-I), z(O)=I. Obviously, for a > I, (2.6) represents the
r-th
factorial moment of the negative binomial distribution with parameters I f now
a < I
I < E(x)+1 = a+b
b = 1+ a-I we have from (3.4) for r=O
q = (a-I)/a
or
and
b
k
that
Then (2.7) becomes I-a > I.
-
257 -
I
(I_a)r(~ _I)(r)
(2.8)
I-a
o
Therefore, the distribution of i.e. =0
there exists for every
an integer
>
r
m.
:5 r :5
[~l l-aJ
-I
otherwise
denotes the integral part of
[w}
where
o
for
m
>
X
w. is terminating,
such that
0
P[ X=r} =
Then, we have from (2.6) for
E(x(m+I»-[(a-l)m+a+b-I]E(x(m»
=
r=~
0
which implies that (a-I )m+a+b-I
0
or equivalently b I-a -I
(2.9)
m
b I-a
which implies that
is a positive integer.
Hence(2.8) represents the
rth factorial moment of
the binomial distribution with parameters and
= ~ I-a
n
-I
p=l-a.
Note.
It can be seen from (2.9) that when
X
is
bounded
I~a > which (since iff
a
<
I.
0
b
>
0)
implies
a
<
I.
Hence
X
is bounded
This shows that the class of distributions
characterized by (2.5) contains precisely the Poisson, binomial and negative binomial distributions.
-
258 -
LAHA and LUKACS [2J provided characterizations of the Poisson, binomial and negative binomial among other distributions by the quadratic regression of the statistic
Q
n
n
~
~
i= 1 j= 1
on
=
S
n a . . X.X. + ~]
~
]
~
j=1
b.X. ]
]
nX.
Since all the distributions they have got are uniquely determined by their moments their result can alternatively be obtained by a method analogous to that of the previous theorem. This is so, because under their assumptions concerning the finiteness of the second moment and the validity of the regression equation, the distributions have all their moments to be finite;
this implies
that they satisfy certain recurrence equations which will lead us to the moments of the distributions in question. To some extent, our results bear also an analogy to those obtained by SHANBHAG [4J. By
Theore~
2.2 the univariate Poisson distribution
has been characterized.
It is of interest now to examine
whether similar characterizations can be derived for generalized Poisson distributions, i.e. with p.g.f. and
g(s)
of the form
for distributions
exp{A(g(s)-I)}, where
>
A
0
valid p.g.f.
Specifically, we turn our attention to the particg(s) = A1 (S-I)+A 2 (s2_ 1 ) The distribution defined by
ular case where
(2. 10)
i~
G(s)
CAL> 0, i=I,2).
i
= 1 ,2)
known in the literature as the univariate Hermite
distribution and was introduced by C.D. KEMP and A.W. - 259 -
KEMP [IJ.
It is a special case of the Poisson-binomial
distribution
(n=2)
and may be regarded as either the
distribution of the sum of two dependent Poisson variables or that of the sun of a Poisson and an independent Poisson "doublet" variable. The following theorem provides a chracteristic property for this form of generalized Poisson distribution. THEOREM 2.3. Let
X,Y
be as in Theorem 2.1.
Then
the condition
(a
holds i f f eters
X
and
a
A.W. KEMP [IJ) that if
[(x{r»
-
=
a2
PROOF. Necessity.
('.I')
0, b
<
0; r=O, 1 ,2 , ..• )
has the Hermite distribution with paramb
a l
> I
~
•
It has been shown (C.D. KEMP and X
is Hermite
(a l ,a 2 )
then
{'a,)rl'H~[ {'a,)-l- a ('a,) --l-] +
l
(r = 0 , 1 , 2 , ••• )
where [n12] H" (x)
n
nlxn-2j
E j=O
.
(n=0,1,2, ••• ;H~(X)=I).
(n-2j)ljI2]
Moreover
(r=I,2, ••• ).
-
260 -
Combining (2.3).
[eyer»~
(2.12) and (2.13) we find that
satisfies a relationship of the form (2.11) with a b
<
(2a 2 )
-I
and
b ... -a l (2a 2 )
-I
• Obviously
a
>
0
and
O. Sufficiency. From (2.3) it follows that
(2.11) holds
iff
(r-I.2 .... ). i.e. iff
(r- I .2 •••• ) .
But this is the recurrence relationship that the factorial moments of the Hermite distribution with parameters -b/a
and
1/2a
Note.
The Poisson "doublet" distribution or the
(p(X-2r)=e
distribution - 0.1 •...
»
if we allow
satisfy. Hence the result.
-A
r
A /r!. P(X-2r+l) - O.
r-
can also be characterized by Theorem 2.3 b
to take on the value
O.
3. CHARACTERIZATION OF SOME BIVARIATE DISTRIBUTIONS WITH INDEPENDENT COMPONENTS We now turn to the problem of providing characterizations for bivariate versions of the distributions examined in the previous section. We first consider the simplest case of having a bivariate form with independent marginals. In what follows a bivariate distribution whose marginals are independent and of the same form will be called "double" (e.g. double Poisson). -
261 -
Indeed, by arguments which are analogous to those used in Secion 2 the following theorems can be proved to hold for double distributions. THEOREM 3. I
(characterization of the double geomet·
-
(ZI,Z2)
= (X I ,X 2 ), K = (YI'Y2)' Z =
~
ric distribution). Let
be random vectors with non-negative integer-
-valued components. Assume that
(3. I)
Then the conditions
(3.2) c
E(x(r)x(R.)
2
I
2
(r-I,2, ... ; 1=1,2, ... ; c l ,c 2 are necessary and sufficient for
X
geometric distribution with parameters
> I)
to have the double -I
cI
-I
,c 2 •
THEOREM 3.2 (characterization of the double Poisson binomial andnegative binomial distributions). Let
(X I ,X 2 ), K - (y l ,Y 2 ) and Z = (ZI,Z2) Theorem 3.1. Then the conditions
=
X
=
be as in
(3.3)
(r-I,2, •.. ; 1 ... 1,2 ••.. ; ai,b i -
262 -
> O.
i=I,2)
X
are necessary and sufficient for
to have one of the
distributions double Poisson with parameters (bl.b Z ) (i=I.Z).
(i)
a.~ =1.
(ii)
double binomial with parameters
-I+b./I-a .•
n.
~
~
(iii)
~
a.
if
~
<
if
p.=I-a .• ~
~
(i=I.Z).
double negative binomial with parameters
(a.-I)/a. ~ ~
and
= I+b./(a.-I) ~ ~
k.
~
i f a;~ > I (i=I.Z).
An immediate consequence of Theorem 3,Z is the following theorem which enables us to characterize bivariate distributions whose marginals are not necessarily of the same form. THEOREM 3.3. Let (ZI'ZZ)
I-a. )
X.
~
for
J
= (XI.X Z ). ! = (YI.Y Z )'
be as in Theorem 3.1.
(3.3) hold i f f (i)
!
P(! =
'"
~)
~
<
a. = I. a j ~
Then the conditions
= P(X I = xl)P(X Z
Poisson (b.). I.
! =
xZ)
where
x. '" binomial (-I+b./I-a.;
J (i:lj;
]
]
i,j=I.Z),
a.: b.
(i i)
(aj-I)/a j )
X.
~
'"
for
Poisson (b . ) •
Xj '" ne g. bin. (I + ~ (Hj; i,j=I,Z), a.=I.a.> I J
~
~
b. a. I J_I;~) (I + ___
a .-
a .
]
for
a.
~
]
b.
(-1+ ---I ~ ; -a.
(iii) X.'" binomial
<
~
I,
a. > J
I;
]
I-a.), X.'" neg. bin. ]
~
(i:lj;
i,j=I,Z).
THEOREM 3.4 (characterization of the double Hermite). Let X (XI.X Z ). ! = (YI.Y Z )' as in Theorem 3. I . Then the conditions
! = (ZI'ZZ) be
(3.4)
(a i
> 0, b i
<
0; i=I.Z; r=O.I,Z ••.. ; t=O,I.Z •... )
- Z63 -
hold i f f
has the double Hermite distribution with
X
parameters
-bl/a l ,
I/Za l , -bZ/a Z '
I/Za Z '
4. CHARACTERIZATION OF SOME BIVARIATE DISCRETE DISTRIBUTIONS WITH DEPENDENT COMPONENTS Let us now consider the problem of characterizing dependent forms of bivariate distributions. We restrict ourselves to the case of the bivariate binomial and bivariate negative binomial with p.g.f. 's of the form n
k
-k
(PII+PIOs+POlt) and P II (I-PIOs-POlt) respectively. A change in the characterizing conditions (3.8) is necessary as it is seen in the following theorem.
THEOREM 4. I
(characterization of the bivariate ~
binomial and negative binomial). Let
K=
(yl,Y Z )' conditions
! = (ZI'ZZ)
=
(XI,X Z )'
be as in Theorem 3.1. Then the
(4. I )
(r-I , Z, ••• ; t-I,Z, ... ; a.,b. ~
~
>
0,
a~¢I, ~
i=I,Z;
bZ
b l
h)
~ )
are necessarg and sufficient for
X
to have one of
the
distributions (i)
bivariate binomial with parameters
n--h-I,
(i"I,Z), (ii) bivariate negative binomial with parameters k-h+l, PIO·(al-I)/(al+aZ-I), P OI -(a 2 -1)/(a)+a 2 -1) ai > I (i-1,2).
-
264 -
for
PROOF. Necessity follows immediately. Sufficiency. From (1.2) for
k=m=1
we have that
the conditions (4.1) hold iff
(r" I ,2, •.• ;
1- I ,2, ••• ) •
Le. iff
(4.2)
(r=O, I .2 , ••• ; 1-0, I .2 , ••• ) • The solution is given by
(4.3)
(r-0,1,2, ••• ; 1-0,1,2). a. <
In the case
~
(i-I,2)
(4.3) represents the
(r,1)-factorial moment of the bivariate negative binomial with parameters
k-h+1
POI=(a2-1)/(al+a2-1). Assume now that we have for
r-i-O
and
PIO=(a l -I)/(a l +a 2 -1),
> 1 (i-I ,2). Then from (4.2) 1 < E(x.)+1 - b.+a. (i-I,2) ~
- 265 -
~
~
iff
-h
>
1.
Then (4.3) becomes
=
(4.4)
I
(I-a )I(I_a )~(_h_I)(I+~) I 2 (0 ~
o
That is when
<
a.
~
I
~ [
I
-h] -I, 0
~ ~ ~
[ -h] -I) ,
otherwise.
x
the distribution of
(i=I,2)
is
terminating Le. there exists a vector
m = (m l ,m 2 ) with non-negative integer-valued components such that P(~
x2
~)
= ~
whenever
= 0
xI
~
and also whenever
ml+1
m2 +1. Then, we have from (3.10) for (m . + I)
E(x. ~
I=m l ,
~=m2
(m .)
(m.)
x. ] )+(I-a.)(m l +m 2 +h+I)E(x. ] ~ ~
~
~
(m.)
x. ] )=0 ]
(i';'j;i,j=I,2) which because
I - a .';'0 ~
(i = I ,2)
implies that
i. e. (4.5) This shows that
-h-I
is a positive integer.
Then (4.4) represents the
(I,~)-factorial
moment
of the bivariate binomial distribution with parameters
n=-h-I
and
PIO=I-a l , P OI =I-a 2 . Hence the theorem is
established. Note 1.
The relationship (4.5) tells something more
It shows that when
X
has a terminating distribution
then
- 266 -
b.
~
a:-:T
<
0
(i=I,2)
~
which since
b.
~
(i=I,2). Hence a.
~
<
> X
0
(i=I,2)
a.
~
<
I
has a terminating distribution iff
(i=I,2).
Moreover, since (i=I,2)
a 2 -1
implies that
/(a l -I)=b 2 /(a 2 -1) and it follows that the differences ai-I b l
b.
~
>
0
and
have the same sign. Hence the class of distributions
chracterized by the conditions (4.1) contains precisely the bivariate binomial and negative binomial distributions. Note 2. I,
If we allow
a.
~
(i"I,2)
to take the value
then the conditions (4.1) reduce to the characterizing
...
conditions of the double Poisson (Theorem 3.2). REFERENCES [IJ
C.D. Kemp - A.W. Kemp, Some properties of the "Hermite" distribution, Biometrika, 52(1965), 381-394.
[2J
R.G. Laha - E. Lukacs, On a problem connected with quadratic regression, Biometrika,47(1960), 335-343.
[3J
E. Parzen, Stochastic processes, Holden-Day, U.S.A, 1962.
[4J
D.N. Shanbhag, An extension of Lukacs's result, Proc. Camb. Phil. Soc., 69(1971), 301-303.
Miss
~vdokia
Xekalaki
J. Panaretos
18 Paxon St.
8 Cratesicleias St.
Athens 812, Greece
Athens 504. Greece
- 267 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION UETHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY), 1977.
ON THE ASYMPTOTIC BEHAVIOUR OF THE GENERALIZED BINOMIAL DISTRIBUTIONS GY. PAP
INTRODUCTION It is well-known that a sequence of binomial distributions can converge to Poissonian or normal distribution. The binomial distribution has the following evident generalization. Let us consider an experiment having two possible outcomes, the events
A
and
A. Suppose that
the probability of the events
A
and
A
may change
from experiment to experiment, namely in the experiment
P(A)
=
j-th
a .• Let the random variable J
the number of occurences of the event experiments. This random variable
n
A
n
in the
be m
is called gen-
eralized binomial distributed random variable with
parameters
{a l ' a 2 , .•• , am} .
B. GYIRES got results about its convergence to the Poissonian and normal distribution. The author proved the existence of other diacrete limit distributions which can be considered as generalized binomial distributions with infinitely many parameters, and there is no - 269 -
more limit distribution. The order of their characteristic and generating function is investigated too, the first in second in
the
[1,00],
We can say that these distributions
[0, I].
are between the binomial and the Poissonian distributions. PRELIMINARIES It is useful to take the generalized binomial distributions by the following scheme. Let the random ~j
variable in the Then
j-th experiment the event {~j}
P(~ .=1)
respectively, or
A
if
occurred.
A
are independent random variables, and
= a.,
]
o
or
take the value
]
P(~ .=0) ]
=
n = ~1+~2+·· .+~m·
and
I-a., ]
Since the characteristic function of ~J' it l+a.(e -I), so the c.L of n is
is
]
=
m
II [ I +a . (e
j=1
it
-I)].
The mean and variance of
n
are
]
m 1:
a.,
j= I
]
E(n)
2 D (n) =
m 1: a . ( I -a .) • j= I
]
]
The binomial distribution has the following asymptotic behaviour. Let the distribution of the random ~m
variable
be binomial with parameter
y
m
(m=I,2, ••• ),
lim mp = ~ < 00 exists, then the sequence m m- oo converges to the Poissonian distribution with
if the limit {~m}
parameter
~.
o < P <
then the distribution of
I,
If the
limit
lim Pm
=
P
t~e
exists and standardized
sequence converges to the standard nermal distribution. Consider now a sequence of generalized binomial distributed random variables (m)
(m)
(m)
(m)
, ••• ,a m } (0 ~ a j {a l ,a 2 the following two results: I.
nm ~
with parameters I).
B. GYIRES proved
The distribution of the sequence
{n m}
to the Poissonian distribution with parameter -
270 -
converges ~
if and
only if the following two conditions are satisfied: m
(m) a.
lim L m- co j=1
2.
(m)
If
A <
]
is independent of index
a. ]
and the limit
lim a. j_co
]
=
exists
a
<
(0
a
distribution of the standardized sequence
m
<
:
1)
n ,', m
(m) aj
=
aj ,
then the tends to
the standard normal distribution. More generally,
it is
true even if m
L a~m)
j= 1 ]
lim
j= 1 ]
(B.
O.
m
L a~m) (l_a~m))13/2
[
]
GYIRES proved these results for the case of gen-
eralized multinomial distributions.) ABOUT THE OTHER LIMIT DISTRIBUTIONS THEOREM I. of index
La.
m:
(m)
and
a. ]
lim E(n
j= 1 ]
(m)
If the parameters
m
) <
co
a. ]
f 0,
a. ]
are independent
and the condition
is satisfied,
then the sequence
of generalized binomial distributed random variables {n m }
{a 1 ,a 2 , ••• ,am} converges in to a discrete distribution.
with parameters
distribution
PROOF. The method of the generating function can be used.
The g. f.
of
m
is
G (z) m
=
n [
j= 1
1 + a . (z - 1 ) 1. ]
the basis of the theory of infinite products,
-
271
-
On
the condi-
tion
<
l: a. j_1 1
implies that the sequence
00
{G
m
(z)}
converges on the whole complex plane to an entire function
So the sequence of
G(z).
nm
tion to a discrete random variable
converges in distribun.(Since the g.f. of
the limit distribution is an entire function,
so
n
has
analitical characteristic function, which is moreover entire function,
too.
For example, if the parameters are 00
then
=1,2, . . . ) 00 2 u II (1- --) j=1 j2
G(z)
II (1+ z-I) j=1 j2
a. 1
sin 1l
I
-:2
1l1f=Z
v' 1- z
(j=
1
since
sinllu The limit distribution can be llU considered as a generalized binomial distribution with a
infinitely many parameters
Pk
a
P(n=k)
{a l ,a 2 , ••• }. The distribution
can be explicitely expressed by the help
of the parameters
{a l ,a 2 , ••• }. We can use the theorem
of Weierstrass stating that if the sequence of entire functions
f
m
(u)
uniformly converges to the function
feu), then the coefficients of the Taylor-series of f
m
(u)
converge to the coefficients of the Taylor-series
feu). Now m m II [1+a.(z-I)]=I+(z-l) l: a.+ j_1 1 j=1 1
G (z) m
l:
+ (z_I)2
ISj I <j2 Sm
so
a. a. + ••• 1 I 12
00 2 G(z)=I+(z-J) l: a .+(z-I) l: a. a. + ••• j=1 1 ISjl<j2 J I J 2
From this
Pk
'"
-
272 -
A
n
:r
a,
a,
\Sj\<j2< ••• <jm ]\ ]2
It is possible to prove that there are no more limit distributions.
nm
THEOREM 2. If the sequence of (m)
eters
(m) (m) ,a 2 , ••• ,am }
{a \
n,
to the random variable m
lim
:r a~m) =
m-~ j=\
]
with the param-
converges in distribution and the condition
< +~ is satisfied, then the m has generalized binomial distribu-
lim E(n )
m-~
n
random variable
tion with finite or infinite parameter or Poissonian distribution or their convolution.
uence of
G (z)
m
~
m-~ j= \
Izl
S
Since
\.
a~m) < +~, there is a bound ]
inequality
~ a~m) <
K
holds for all
m, so because of
]
(m)
m
IGm(z)1
m
S
n
e
~ a,] Ia,(m) ( z _\) I Iz-\ I '_I ]
-e]-
jo:\
Izi <
for So the sequence
Izl <
for which the
K
m
j= \
disk
n, so the seq-
is convergent on the unit disk of the
m
complex plane, that is for lim
d
nm
PROOF. Since we assumed that
R
{G (z)} m
R.
is uniformly bounded on the
for arbitrary
>
R
O.
Now we can apply the theorem of Vitali stating that if the complex functions
f
n
(n)
and have a common bound on the domain
are homolomorphic T,
{f (u)} sequences is convergent on a set n has an accumulation point in the inside of
- 273 -
and the HeT T,
which then the
is convergent on the whole domain
T,
moreover the convergence is uniform in the inside of
T,
{f
sequence
n
(u)}
and the limit function is also holomorphic on Now the domain
H
is the unit disk.
Iz I <
is the disk
T
R
Since
R,
T.
the subset
is arbitrary the limit
function is an entire function. It is a well-known corollary of the theorem of Rouche that if the complex functions the domain
f
n
and the sequence
T,
iformly in the inside of
are holomorphic in
(u)
{f
n
converges un-
(u)}
to the function
T
f(u),
then
for arbitrary simple, closed curve in the inside of
T,
f(u) and f
(u) have the same number n of zeros in the inside of the curve, taking into and for 1 arge n
consideration their multiplicity, too. This implies that the zeros of
f
n
(u)
G (z)= ~ [I+a~m)(z-I)] m j= 1 ]
Now the zeros of (m)
z. ]
=
1 1--(m)E(-co,O], a .
f(u).
accumulate to the zeros of
G(z)
so the zeros of
are are also
]
G (z)
real and non-positive, if the zeros of ulate at all. -
If the parameters
taking into consideration
m
(m)
(m)
{a l
,a 2
accum(m)
, ••• ,a m
their multiplicity -
}
accum-
ulate to {a .J, then on the basis of the infinite product ]
representation of entire functions, generator function is h(z)
the form of the
G(z)=eh(z)O [I+a .(z-I)], j ]
is an entire function.
By the help of the expan-
sion into series G (z) m 10 g ""O,...,[O-:I,.::;+:....a-...,..(z---:'I~)] j
(z-I) [
~ a ~m) - ~ a.J
j=1 ]
]
-
where
274 -
j
]
+
- r j
m
=
h(z)
thus
A
A(z-I), where
m
lim r a~m) m- oo j=1 ]
a.E[O,oo). So the general limit distribution is a ]
convolution of a Possionian and a generalized binomial distribution with finitely or infinitely many parameters. ABOUT THE ORDER OF THE C.F. AND F.G. OF THE LIMIT DISTRIBUTION We know that the order of an entire function may be
o
S
p S +00, the order of a characteristic function is
at least I, except if it is the constant c.f. The order of the c.f. of the binomial and Poissonian and
distribution are
+00, respectively. It is poss-
ib1e to prove that the order of the c.f. of generalized binomial distribution with infinitely many parameters may be artbitrary number between the parameters are
<
I,
and
Y
>
an=q
nY
(n=I,2, ••• )
0, then the order is
case of the geometrical series as the order of c. f.
+00, namely i f
I and
{qn}
where
I I + Y
. So
<
0
q
<
in the
the order is
2
of normal distribution. This can be
proved by the help of estimation of number of zeros of the c.f. in the disk (n=I,2, ••• )
Izl S R. In the case
the order is
a
I
n
"2 n
+00.
The order of an entire generating function may be
o
S
p S +00. The order of g.f. of binomial and Poissonian
distribution are
0
and
I,
respectively. In the case
of the generalized binomial distribution with infinitely many parameters the order is at most disk of i.s
I, because on the
Izl S R : IG(z)1 S e(R+I)K, so the maximum modulus G(z) : log M(R) S (R+I)K, thus the order of G(z) p S lim sup 10g(R+J)K 10gR
I.
- 275 -
We can prove that this order may be arbitrary number
n
00
-I
log
-2
for example if the parameters are
I ,
{n -I h}
or
{q } n= I
{n
and
0
between
where
00
n=1
<
0
y
<
I
or
00
n}n=2' then the order of g. f. of the correspond-
ing distribution are
0, y
and
I,
respectively. The
proof is based on the followiRg theorem: Denote by
r l ,r 2 , ••• the moduli of the complex zl,z2"" • Let a be the infimum of the set
numbers
for which the series
of powers
~
n=1 assume that
<
0 S a
I.
r
-a n
converges,
Then the order of the entire
00
n
cp(z)
function
n=1
(1- -=-) z n
true also for the case of
is
a. The statement is 00
0=1, if the series
~
r
-a
n=1 n converges for
a-I, too.
REFERENCES [IJ
B.
Gyires, On the asymptotic behaviour of the
generalized multinomial distributions,
(to appear).
[2J
E. Hille, Analitic function theory, Vol. 2.,
1962.
[3J
S.B. Holland, Introduction to the theory of entire functions, Academic Press, New York and London,
1973. Gy. Pap Mathematical Institute of the Kossuth tajos University 4010 Debrecen, Pf.
18, Hungary
-
276 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. AnALYTIC FUNCTION HETHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
ON THE STRONG MARKOV PROPERTY OF THE EXPONENTIAL LAWS* B. RAMACHANDRAN
The contents of this paper constitute yet another illustration of the power of the methods used by the late Academician Ju. V. LINNIK in his monumental study of the implications of the identical distribution of two linear statistics in independent and identically distributed random variables, with special reference to the common distribution of the latter being normal (LINNIK,
[5J). SHIMIZU
[9J and RAMACHANDRAN and RAO [8J
are two other instances of the exploitation of these methods of Linnik.
(See also the Remark at the end of
this paper.) Let
X
and
y
be independent non-negative random
variables on some probability space, and let denote their distribution functions and
*
PG
F
(d.f. 's), and
and
G
PF
their probability distributions - equivalently,
The author's attention was drawn to the problem considered in this paper by Mr. ANANT P. GOD BOLE and Dr. Z. GOVINDARAJULU. -
277 -
the probability measures induced by (RI,B I ).
tivelyon
If
and
F
G
respec-
has an exponential distribu-
X
tion, then it is well-known that the following relation holds, whatever be the distribution of
P (X >
P(x> y+t)
(I)
y)
Y:
P (x > t)
(t
>
0).
This relation may equivalently be written in the form
J (I-F(y+t»-(I-F(t»(I-F(y»dG(y)
0
o or co
for all in
o
J (F(y+t)-F(y»-F(t)(I-F(y»dG(y) o
(2)
t
>
t
O. The integrand in (2) is zero identically y ~ 0
and
if
is an exponential law. However,
F
relation (2) is also satisfied for instance if
I-F(n)=pn
"geometric" d.£. with integers
n, for some
is a
F
for all non-negative
pE(O,I), and
G
is any d.f. with
support on the set of non-negative integers, the integrand in (2) being then identically zero in whenever
y
t
~
0
takes on any non-negative integer value.
It is then natural to raise the question: what can be said about
F
if (2) holds for some
particular, when is
F
G, and, in
necessarily exponential? This
problem was considered briefly in KRISHNAJI [4J. We begin with some preliminary remarks by way of disposing of the less interesting cases first. first that if
=
I,
F
and
G
are such that F
and
example would be: F=Oa' G=ob' with
°
x
PG({X:F(x)=I})=
then there is nothing further to be said,
being satisfied for all such
278 -
(2)
G: an elementary 0 < a < b, where
is the degenerate d.f. with its jump at -
Note
x.
In what
follows, we shall assume therefore that (3)
Secondly,
In
F(x)
PG({x:
the case
only conclusion that
F(O)=O
unles
<
J.
is of no interest, the
G=oO
(2) yields in that case being that
F=OO'
We finally remark that if
(2) is satisfied, then
we must necessarily have
(4)
O})
For,
<
I.
if the above were to be an equality, then (2)
implies that
J [F(y+t)-F(t)] dG(y)
0
o for all where
> 0, so that, for every
t
A
t
possible cases: either (a) G(x) so that, n,
for every
PG[(n,
t
> 0
+oo)nAt]=PG[(n,
y =y (t)
>
n
such that
letting
n -
00, we have
n
n
for some
> 0,
t
{y : F(y+t)=F(t)}. Now consider the two
=
x
>
0,
the support of
< I
for every
>
0,
and every positive integer +00)1
>
0
and so there exists
y EA , i.e., F(t+y )=F(t), and
n t F(t)=1
n
or
F=OO;
in which case, let G.
x
B=O
or
(b) G(x)=1
the maximum of
B
corresponds to
G=oO
already
B > O. Then
PG {(B/2, B]nAt} = = PG {(B/2, Bl} > 0, so that, for every t > 0, there considered. Let then
exists
Yt
> B/2
Thus,
F(Yt+t)
such that
F(t)=F(t+B/2) for all
t
> 0
F(t), so that
and hence, again, F=OO'
in both cases (a) and (b), we obtain the conclu-
sion that
F=OO'
Hence
(4) must hold if
Thus, ignoring the case:
PG({x
:
(2) does.
F(x)=I})=1
what follows, we have that both the sets: -
279 -
in
{x
= O}
F(X)
and
{x
I}
F(X)
PG-measure less than one, so that
have 00
- J I-F(y)dG(y) <
o
0
<
=
c
I. We_shall in fact assume that*
G(O) < c < I.
(Sa)
Then there exists a unique
A
>
satisfying the rela-
0
tion 00
J
(5b)
e
-AY
dG(y)
c.
o We are now in a position to state our characterization
THEOREM. Let with support on fied.
Let G
F
G ~ 00
and
[0,00)
be d.f.'s on
such that relation
(2)
RI
is satis-
be assumed to have m.g.f., i.e.,
00
J eyadG(y) <
for some
00
>
a
o
proof').
for all
>
t
b)
0,
being given by
A
(5b),
I-F(t) if
= e -At is any
G
with the origin as a lattice point; and
I-F(t)
=
h
is a
for all
Ut). e -At
is given by (5b)
lattice d.f. if
(5a) being assumed satisfied)
(subject to the above restrictions) other than a
lattice d.f.
A
(condition
'Note added in
is necessarily exponential:
F
a)
d.f.
Then
(see
0
and
t
>
0, where again
is periodic, i f
~
G
is a
with the origin as a lattice point. Further "span" for
G,
h
is also a period for
(We recall that a d.f. on - or arithmetic - d.f.
RI
~.
is called a lattice
if it is purely discrete and its
discontinuity points form a subset of a set of the form {a+nh: n
*
If
integer}, where
a
is a real, h
>
0.)
c~G(O), the assertions of Theorem need not hold. -
280 -
COROLLARY. G,
oa
say
then
F
If (2) holds for two degenerate laws
and
0b' with
>
a,b
0
and
alb
irrational
is an exponential law.
REMARK. This corollary is the essential content of MARSAGLIA and TUBILLA [6J, where the property discussed is picturesquely described thus: in addition to being the only d.f. which is memoryless, the exponential has the property of being the onlyd.f. which is "occasionally forgetful".
KRISHNAJI [4J has pointed out that
if (2) holds for all degenerate laws
G, then
F
is
necessarily exponential in consequence of the fact that I-F
then satisfies
the Cauchy functional equation on
the positive real axis. PROOF OF THE COROLLARY. Take linear combination of a
0
and
a
G
to be any convex (2)
holds for such
G, and part (a) of the Theorem is in force.
part (b),
I-F(t)
sl(t)exp(-Alt)
Or, by
as well as
and = s2(t)exp(-A 2 t) for some A I ,A 2 > 0 and sl S2 having periods a and b respectively. Since the Sj' being periodic, are bounded, it follows that
AI=A2
has two periods which are incommensuS(=S I =s 2 ) rable; being right-continuous as well, it is therefore
and
necessarily a constant: s=1
then, from (2).
PROOF OF THE THEOREM. Taking the Laplace Transform (L.T.) of both sides of relation (2), we have for Re z
>
0,
f (I-F(y+t))-(I-F(t))(I-F(y))dG(y)dt=O. o Let
h
be the L.T. of
I-F, i.e. -
281
-
J e-tz(I-F(t»dt o
h(z) Then, for
0
< Re
Z
<
(Re
Z
>
a,
J J (l-F(y+t»e-tzdt dG(y)=
h(z) J I-F(y)dG(y)
o
o
0
00
y
0
0
h(z) J eyzdG(y)- Je Yz J(I-F(u»e
o h(z)
noting that
0).
is defined for
Re
>
Z
-uz
0
du dG(y) and the
integrals in the above relation are defined for
Re z
<
by assumption. Recalling the definition of the number
c
by relation (Sa), we have
h(z) (f eyzdG(y)-c) =
o
f e Yz J[ I-F(u)] e-uzdu 0
0
(0
< Re
dG(y)
or (6)
h (z)
K(z) o(z)
Z
<
a)
where y
K(z) (7)
J e Yz J (I-F(u»e-uzdu dG(y);
o
0
o(Z)
K(z)
and
h(z)
in
of
K(z)
of
o (z) •
o(z) Re z
are both analytic in
>
O. Relation (6)
in the region
0
Re z
<
a
and
shows that the zeros
< Re z < a
cancel out those
These relations suggest the applicability of the kind of analysis used in LINNIK [SJ, SHIMIZU [9J and RAMACHANDRAN and RAO [8J.
In KAGAU, LINNIK and RAO [3J
a simplified version of Linnik's approach, due to -
282 -
a,
A.A.
ZINGER, is presented, but, contrary to the impres-
sion
one is
likely to obtain from the presentation
(vide p. 49) there, the scope of the arguments there is not confined to the case where periodic analytic function.
is an almost
a(z)
It is therefore necessary
for us to go back to the fundamental paper of Linnik's for some of our arguments (the page numbers cited below refer to the English translation [5J of the Russian original). We begin with two basic facts. Taking discs with centres at the points on the vertical line
Rez=-N, where
sufficiently large, and of radius
<
and using the fact that
a),
N+r
N(a,b)
<
(0
>
0
is
r=r(a,b)
<
is bounded in every
a(z)
Re z S 8 «
half-plane of the form
=
N
-N+i(y+I/2)
and applying
a)
Jensen's theorem (cf. TITCHMARSH [IIJ, p.
125), we see
that LEMMA I. The number of zeros of S
y+ I,
is bounded by a number
depend on
in any closed
a(z)
« a),
a S Re z S b
rectangle of the form
S
y
Im z S
which does not
n(a,b)
y.
It is clear how depending on
a
Nand
and
r
b.
Again using the fact that every half-plane
have to be chosen is bounded in
a(z)
Re z S 8, with
8
<
a,
and Lemma I,
we conclude that LEMMA 2. Given > 0
such that
the strip: radius
e:
y > 0,
I a (z) I
e: > 0,
> m(y,e:)
there exists
for all
z
m(-y,e:)>
lying in
-y+e: S Re z S a-e:, but outside of discs of
with centres at che zeros of
a(z).
A proof from first principles of this lemma muy be modelled on that of Lemma IV in Linnik [5J, pp.II-12. -
283 -
since
I-F
co
!(I-F(t»e-
A
tdt
o (I-F(t) )e
of Re z
> -A
is bounded, the integral exists for all
-At
is
>
A
0; also, the L.T.
h(z+A), with the half-plane
being contained in its half-plane of conver-
gence. Hence. by a complex inversion formula for the L. T.
(cf. DOETSCH [2J,Theorem 27.1, p.
179) - also vide
[3J, relations (2.2.16), p. 50, which, however, have to be revised along the lines indicated below so far as the range of values of we have, for all
and
x"
>
T
below are concerned -
x'
Principal Value)
O,(PV
x"+i oo
T
! e-At(I-F(t»dt
PV
o
e f 2ni x "-ioo
>
(x"
which is obviously (x'
> A
x'+i oo
PV 2ni
f
e
0
TZ
h(z+fI) dz z
arbitrary),
arbitrary)
T(z-fl)
h(z) -;::::r::-
dz ,
x'-i oo
which, by a simple application of Cauchy's theorem on residues, is x+i oo
=
PV 2ni
f
e
T(z-A)
h(z) -;::::r::-
dz+h(A)
x-i oo (0
<
x
<
A
arbitrary),
so that co
(8)
f e-
A t[ 1- F (t)] d t
=
T
= -PV
x+ioo _1-
f 2ni x-ioo (0 -
284 -
< x < A arbitrary).
(The final relation above also follows from Theorem 27.2 of DOETSCH [2 J , in view of Re z
>
-A
h(z+A)
being defined in
and thus for some negative real values of z. )
Then, combining (6) and (8) , we may write: co
1 J e- At [ I-F(t)] dt =-21ti lim
(9)
x+iT
J
T-co x-iT
T
<
(0
e 'r(z-A) K ( z ) dz (z-A)a(z)
<
x
min(a,A».
It now follows from relations (5) and (7) that there exists a unique that
>
0
such that
Re z
<
-A. Given
£
>
1m z • ± T
every zero of (B
<
0
and
0, we can on the strength
of Lemma 1 choose a sequence {TmJlines
=
a(-A)
has no zeros, real or complex, in the ha1f-
a(z)
plane
A
such that the
co
are all at a distance
m
a(z)
lying in the strip
>
from
£
-A S Re z S B
a); then using Lemma 2 and arguing for instance as
in [8J we see that, for any fixed
A
>
0
and all
T
>
0,
co
J (l-F(t»e-Atdt
(10)
= lim s
T
m
(A,T)
where s
m
(A, T) •
the sum of the residues of the function {-
eT(z-A)K(Z) (z-A)a(z) }
at those zeros of
a(z)
which lie in the rectangle: -A S Re z S 0; 11m z
I < Tm•
The zero of the denominator of the above function at the point
A
does not enter into our calculations for
obvious reasons, and Re z
<
a(z)
does not have any zeros in
-A. Calling a zero of -
a(z)
285 -
active if the above
function has non-zero residue there, we can establish the following lemmas. LEMMA 3. If a2
then
a1
and
inf {Re w: w
is an active zero},
sup{Re w: w
is an active zero},
are themselves active zeros.
a2
The proof of this lemma depends (only) on the fact [1-F(t)]e- At with h(z+A) as its
that the function
L.T. is non-negative, and may be conducted along the lines of those of Theorems 2.3.1 and 2.3.2 of [3], our argument being in fact shorter since we need and claim fewer facts. As already pointed out, a(z) zero, namely
=
has only one real
-A, so that Lemma 3 implies that
-A. In other words, all the active zeros of
on the vertical line
Re z
attention to the zeros of
=
a 1 =a 2 =
a(z)
lie
-A. We therefore turn our
a(z)
on this line, whether
active or not. LEMMA 4.
=
Re z
the line
lattice d.f. however, G
-A
if
has no zeros other than
G
-A
on
is a d.f. other than a
with the origin as a lattice point; if, is such a lattice d.f. with span
the zeros of
h, then
on that line form a set of the form 21t integer}, where vo = 11 . a(z)
{-A+inv O : n (ii)
a(z)
(i)
the zeros of
a(z)
Re z - -A
on the line
are all simple.
PROOF. We have in view of
G
~
a(-A)
=
0, a'(-A) -
00. Also, a(-A+iv)
'"J e-AY(I-cos vy)dG(y) o -
=
286 -
O.
=
'" -YA Jye dG(Y)
o 0
implies that
> 0
If
G
is a d.f. other than a lattice d.f, with the
origin as a lattice point, this relation can be satisfied only for
If
v=O.
G
is a lattice d.f. of the kind
described, then this relation can be satisfied for, and only for, G. Let =
-A
2klt 11 where
of the form
v
2lt
h
is a span for
vo h , so that the zeros of a(z) on Re z = integer. It is easy to are the points -A+inv O ' n
check that a'(-A+inv O ) = J ye-YAdG(y)
>
0,
o
so that again in this case also the zeros of the line
Re z
=
-A
a(z)
on
are all simple.
Continuing with the proof of our main result, let us consider the two possible cases: Case 1. G
described. Then
is not a lattice d.f. of the kind
-A
is the only relevant zero of
a(z)
and we have from (10) that 00
J e-
A =
t(I-F(t»dt
T
res
{_ e
Z=-A
T(z-A) ( ) K z } (z-A)a(z)
Differentiating both sides with respect to e
-AT
[ I-F(T)]
T. we have
e' A
or I-F(T) = e"
A
and since the left hand side is independent of A. it -TA I-F(T) = e as required to prove.
follows that
-
287 -
Case 2. G
In this
case~
is a lattice d.f.
of the kind described.
we see that
J e-At(I-F(t))dt = lim T
('
lim e- A+ N-""
A)
N
T( ~ c A
e
inTvO
).
,n
-N
Denoting the limit of the expression in parentheses by
nA(T), we see that
is differentiable and periodic
2~ = h. We also have
with period
e
nA
Vo
-T(A+A)t"., AT, ( )
or
Since the left hand side does not depend on follows that
sA=s, or
is periodic with period
I-F(T)
=
s(T).e
-TA
A, it
,where
S
h. This completes the proof of
our theorem. REMARK. It is clear from the foregoing arguments that the condition that
G
has m.g.f. is imposed only
to enable us to exploit the methods used by LINNIK [5] and, following him, by SHIMIZU [9] and RAMACHANDRAN and RAO [8]. In this context, we may refer to a recent paper by DAVIES and SHIMIZU
[I]
as of interest:
it
proves using "real variable" arguments results proved earlier in SHIMIZU [9] and RAMACHANDRAN and RAO [8] using "complex variable" arguments. See the following -
288 -
Note added in proof. The object of this Note is
to point out that the assumption that the m.g.f. of
G
exists can be removed, so that our basic result is true without
pre-conditions on
any
(ignoring
trivial
cases).
G
other than (Sa)
We begin by pointing out
that we need only consider the case where
G(O)
=
0: the
general case can be reduced to such a case by re-writing the basic equation as
f
C*T(t)
T(t+y)dG*(y),
(O,co) G>~
where
T -
tion, 0
G(y)-G(O) \-G(O)
(y)
< co)
c* _ C-G(O) (C > G(O) by assumpand - I-G(O) I). We shall therefore assume in what
\-F
< c* <
follows that
(0 S y
G(O) = O.
Then, conditions (2) and (Sa) being satisfied, and C
and
A
being given by relations (5), we have
f
CT(t)
T(t+y)dG(y).
(O,co) Choose and fix any
a e(O,A), and define
T
a
and
according to at
T (t) =- (I-F(t»e a
G
a
(x)
then, for all ( I I)
=-
C t
,
f (0, x)
> 0,
T (t) =f T (t+y)dG (y) a (O,co) a a
where, obviously, Ga (O)=-O
and
- 289 -
Ga (+co)
> I. We can
G a
therefore choose and fix
a
<
~
b
<
00
G (b)-G (a)
and
ex
ex
and
such that
b
0
<
a
<
I. Then we have from (I I)
that T (t) ex
f
~
T (t+y)dG (y),
(a,bj
ex
ex
whence it follows by contraposition that there exists some
in
!;;=!;;(t)
(a,bj
such that
T
(t+O S T (t).
It
ex ex t SuS t+~, T (u) S T (t)eex~ S is clear that, for ex ex exb t > 0, there S T (t)e ,so that we have: for every ex S T (t), exists t' such that (I) t ' ~ t+a, (2) T (t') ex ex and (3) T (u) S eexb T (t) for t SuS t'. Thus there ex ex such that (a) T (t ) S exists a sequence {t} .. 00 n ex n S T (t )eexb for S Tex(t l ) for all n, and (b) T ex (u) ex n for all tn SuS tn+1 so that Tex(U) S Tex(t l )eexb u
~
t l ,
whence
Tex
is bounded.
Now, to the bounded non-negative function
f
satisfying (II), where
eex Y dG
(y)
<
00,
T
ex the argument
(0,00) ex in the body of the paper can be applied mutatis mutandis T and G) ; leading taking the places of G (with T ex' ex is of one of the two T (t ) us to the conclusion that ex e-]..It is e-]..It~(t) ]..I = I- - Ct. and where ~ or forms:
periodic
, etc. , thus completirig the proof of the
Theorem. REMARKS.
I.
The argument shows that in fact
00
f eexxdF(x) <
o
00
for all
ex
<
1-0
2. Apropos of the Remark made at the end of the
main paper, SHIMIZU [IOJ has given a "real variables proof" of our basic result -
also without any conditions
(such as the existence of m.g.f.) on
Go
The auxiliary
result proved by him may be cite~ here: "If -
290 -
T
is a
non-negative and right-continuous function satisfying
J
T(x) ~
the relation
where
T(x+y)dG(y)
G
is a
[0,00)
d.f. with m.g.f., and sup
T(x+y)
T
S T(x)C(n)
satisfies the growth condition: for all
x,n
~
0,
where
C(O+)=
OSySn
=1,
then
T
is bounded". The proof of this result has
to invoke much more delicate arguments than above view of the total variation of (and not> 1) -
G
in
being precisely one
including the observation that
Tis
lower-semi-continuous. 3. SRANBAG has called attention to the related results due to MEYER [7J, p.
CROQUET and DENY, stated and proved in
152, by martingale arguments.
REFERENCES [IJ
L. Davies - R. Shimizu, On identically distributed linear statistics, Ann. Inst. Statist. Math.,
28(1976), 469-489. [2J
G. Doetsch, Introduction to the theory and application of the Laplace transformation,
translation
1974.
from the German original, Springer-Verlag,
[3J
A.M. Kagan - Ju.V. Linnik - C.R. Rao, Characterization problems in mathematical statistics,
transla-
tion from the Russian original, John Wiley,
[4J
1973.
N. Krishnaji, Note on a characterizing property of the exponential distribution, Ann. Math. Statist.,
42(1971), 361-362. [5J
Ju.V. Linnik, Linear forms and statistical criteria, I and II, Ukrainian Math.
Journal
(1953); English
translation Selected Translations in Mathematical Statistics and Probability, Amer. Math.
Providence, 8(1962).
- 291 -
Soc.,
[6]
G.
M~rsaglia
- A. Tubilla. A note on the "lack of
memory" property of the exponential distribution Ann. Prob., 3(1975). 353-354.
[7]
P.A. Meyer, Probability and potentials, Blaisdell. Waltham. Mass •• 1966.
[8]
B. Ramachandran - C.R. Rao. Solutions of functional equations arising in some regression problems. and a characterization of the Cauchy law. Sankhya A. 32(1970). 1-30.
[9]
R. Shimizu. Characteristic functions satisfying a functional equation I. Ann. Inst. Statist. Math •• 20(1968). 187-209.
[10] R. Shimizu. Solution to a functional equation and its application to some characterization problems. Research Memo. No. 131. The Institute of Statistical Mathematics. Tokyo. [ I I ] E.C. Titchmarsh,
The theory of functions, 2nd ed.,
Oxford Univ. Press. 1939. Note. The fact that if
C = G(O). our theorem need
not hold is shown by the following example due to Dr. J.S. HUANG of Guelph. Canada: take
S x S I
and
G(O) -
I-e
-I
;
G(x)
F(x)" x for -x = I-e for x
B. Ramachandran Indian Statistical Institute 7 SJSS Marg. New Delhi. 110029 India
- 292 -
0 S ~
I.
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. AUALYTIC FUNCTION l-lETHODS IN PROBABILITY THEORY DEBRECEN (HUUGARY),
1977.
ON SOME FUNDAMENTAL LEMMAS OF LINNIK B. RAHACHANDRAN
The first name that springs to mind, when one speaks of the methods of complex analysis in the theory of probability and statistics, is undoubtedly that of the late Academician YU.V. LINNIK. As is well-known, two of the major directions of his work in this area are on: (A) characterization of the normal law through the identical distribution of two linear forms in independent and identically distributed random variables (i.i.d. r.v. 's) specifically, establishing necessary and sufficient conditions for the normality of the common distribution of the latter in terms of the coefficients in the former; and (B) decomposition of probability laws. His work in area (A) was simplified by A.A. ZINGER, and an account of this simplified version may be found,
for instance, in KAGAN,
LINNIK and RAO [3J. The basic idea of using the Laplace transform to solve a certain functional equation, as well as other arguments used there, have also been found useful e1swhere, in widely different contexts: as examples. we may cite: SHIMIZU [8J,
RAMACHANDRAN and RAO [7J -
293 -
and
RAMACHANDRAN [6]. Linnik's work in area (B) was summed up in the book, LINNIK [4], and an extended and expanded
version, with simpler proof due to I.V. OSTROVSKII of many of Linnik's basic results, as also later contributions by other authors, are to be found in LINNIK and OSTROVSKII
[5].
It is the modest aim of the first section of this paper to place on record rigorized proofs of some lemmas stated by LINNIK, which are basic to this investigations in area (A): these may be found quoted as Lemma 2.4.2 in C)].
Relations
(2.4.58) and (2.4.68) there are obviously
insufficient for us to make the desired conclusions. In the second section, we state and discuss a conjecture on power-series suggested by the first of the three basic lemmas of Linnik's in his result on necessary conditions for an infinitely divisible (i.d.) law with normal component to have only i.d. components. I.
PROOF OF SOME BASIC LEMMAS OF LINNIK'S IN THE
CONTEXT OF CHARACTERIZATION OF NORMALITY THROUGH THE IDENTICAL DISTRIBUTION OF TWO LINEAR FORMS IN I. I. D.
R. V. 'S
The lemmas under reference may, as already stated, be found quoted and "proved" as Lemma 2.4.2 in C)]. We shall not consider the details of proof of all the cases to be considered, but shall only prove the following two assertions (Lemmas I. I and 1.2) as typical. LEMMA 1.1. If
[ Y1/2]
YI
is even, then
~
2
([x]
is not an even integer, while equals the largest integer
S x)
exp (-A t
2
-I t 1Y I ) - 294 -
is a ch.
f.
for all large
PROOF. Let Re z
~
O}
f(z)
=
>
A
O.
2 YI exp(-Az -z )
and 00
(I • I)
{zEC I ,
for
00
=
I e-itxf(ltl)dt
2 Ij (x)
2 ReI eitxf(t)dt. 0
_00
An obvious application of Cauchy's theorem (cf.[3J, p.
76) shows that, for any 00
( I .2)
I e
itx
f(t)dt
=
o
I e
A izx
>
0,
f(z)dz+I e
LI
izx
=
f(z)dz
II+I 2
L2
where
o s v
{z=iv
L
=
2
o
{z=u+iA
In what follows,
S A},
S u S oo}.
will denote an absolute pos-
AO
itive constant (i.e., one not depending on and
B
A
or
x)
a constant or a variable quantity which is
bounded by some absolute constant. Neither
AO
nor
B
need always denote one and the same quantity. Let us choose and fix the constants OE(O,I); A
( I .3)
>
2Y I
and
0
and
A(I+o)
A
<
such that 4Y I
and take
0(A)
( I .4)
[A(I+o)A
log A] 1/2
and
We consider separately the cases: (a) x (b) 0 S x
< 0(A),
case for
A
~
and show that
AO. Note that -
295 -
Ij(x)
A log x
A(x)
>
0
x ~
0(A)
in either
and
(i)
(1.5)
decreases on
A (x)
o
S A(x)
< A for
> 0, 0 S
(I .6)
(ii) for
(1. 7)
(iii) for complex
x
e
and
(e, 00)
> I,
x -x
I
-I+x S I
2
x ,
z,
x ~ 0(A),
(a) Let us now consider the case:
A
~
AO
considered large enough; then, in view of (1.5), we have: (1.8)
(iv)
( I .9)
(v)
A[A(x)] 2 S A[A(0(A»] 2S
A(x) _< A(0(A» x 0 (A)
i
log A,
< 1 - 2A
so that (1.10)
(vi) for
0 S v S A(x),
I
I
o.
Av - IX S AA(x) - IX S
Taking
A-A(x)
in (1.2), but writing
A
or
according to convenience, we have S e -xA(x)+AA
(1.11)
S
-
B
IA
2 00
J
o
2
I -
YI
'd
-Au (u+iA), e e
exp (-A10g X+AA2) S
A 1 - (1 + I -A BA4 S x S Bx
1
fi)
- 296 -
-(YI+I)
o (x
)
u <_
A(x)
on account of (1.8), turn to
I
(1.4) and (1.3) respectively. We
I.
=
Re II
A(x)
~
Re i
2 YI exp(-vx+Av -v exp(i
so that, in view of (1.5),
IR e (1.12)
A(x) II- R e
J
~'
o
I
yi»dv
(1.7) and (1.10),
2 Y . ('~ ~ e -vx+Av ( -v 1 exp 2 Y I » dv
I
=
,A(x) -VX+Av 2 YI ,1[ IRe II- Re ~ J e (I-v exp(~ 2 yl»dvl o vx A(x) - T 2Y I A(x) +A 2 2Y I vx S B J ev v dv S B J e v dv S
o
S Bx
o
-(2y l +l) •
YI ,
But, in view of the conditions on
sin
21t
YI
>
0
and hence we have
1t
sin
2
A(x) 2 YI -vx+Av v dv YI J e A(x) -vx YI e v dv"
1t
~
sin
2 Y I 0J
=
sin
2
1t
~
0
( I. 13)
00
YI
(J e
o
-vx -Y I v dv-
00
J
-
e vXv
YI
dv).
A(x)
The second integral on the right hand side is = BX-A(I-a) for any a >
S B exp[-(I-a)xA(x)]
A
>
2y l , a
that
A(I-a)
o.
Since
may be deemed chosen sMall enough to ensure
>
(1.13) is, for
l+y l , so that the right hand side of x
~
SeA), A
~
AO'
(I • 14)
- 297 -
s
From (1.11)-(1.14), it follows that
x
~
!I(x)
>
0
for
~.AO.
S(A), A
Turning to the case
0
<
x S S(A), we have in view
of (1.6) that 2
GO.
IRe f e~UX-AU (e- u
YI
Y
-I+u I)dul S
o (1.15) GO
S
f
e
2 2Y I -Au u
-(y
du
+ I
BA
I 2)
0
Now, GO
(1.16)
Re f e
iux-Au
2
I --- e
du
212A
0
x 2A
~
and, setting
2 x - 4A
2
Re
j
eiUX-AU2uYldU
o (1.17)
Re{i
GO
f
o
2
yl+1
x GO 2 Y e 4A f eA(V-~) v I dv } + ---
2 0 -~ GO
Y +1
+ Re {i
Re e
x - 4A
I
e
4A
2
f e -Au (u + i 0
o
Y
I
d u}
by an obvious application of Cauchy's theorem. Under the conditions imposed on
YI '
the first term on the right
is negative, and the absolute value of the second term is
- 298 -
B
Yl t2 ) A
(...!....)
and, noting that
Y
2A
I
the
absolute value of the second term on the RHS of (1.17) is 2
x I a(A- e- 4A ),
the
a-notation applying to I
A
-I
e
x2 -4A
~ A
-I
( I . 18)
-(-4IA (I+O)+ A
since
A(I+o)
< 4Y I
~
i)
~ A
-(Y + I
!) 2
>
y(x)
0
for
0 S x S 0(A)
as well,
x2
I
212A
exp(- 4A)' proving the lemma.
LEMMA 1.2. If teger and
I
exp(- 4A(I+O)log A)
as per (1.3), and it follows from
(1.15)-(1.18) that being
A. Finally,
[Y I /2]
<
2 S YI is odd,
Y2' Y I
then,
is not an even in-
for all
A ~ AO'
the
functions 2
-It I
(i)
exp(-At
(ii)
exp(-At -a O
YI
log
Itl)
and
2
It! YI -It I Y2 )
are ch. f. ' s. PROOF.
(i)
(a) x
~
0(A)
:
I2
ier, and, for an absolute constant - 299 -
is estimated as earlMI , we have
(1.19)
For
M2
>
to be chosen below, let us suppose that
0
is so large that, for
(1.20)
vi >
110g
~
x
~
0(A), A
max(M 2 ,R)
<
0
for
AO'
<
v
A(x).
Then, by (I. 7),
IRe
A(x)
I 1 -Re
.
f
~
S B
o
2
r--
Y
R e -vx+Av ( -v 1 exp (·1/ ~V2 + 1 og v » dv I < -
A(x) 2Y 1 f v (110g vl+
o and, for any fixed (I .21)
S B
a
>
I)
2
2
e- Vx +Av dv
0, this is
A(x) -~ 2y -a f e 2 v I dv S
-(2y +\-a) Bx
\
o YI , cos
Under the assumption on
2R YI < 0,
and, taking
(1.20) into account, we have A (x)
. f
Re
~
o ~
e -vx+A v
R
I cos 2
-I
2
Y
( -v I exp (.~
2R Y I)( 1 ogv+~. 2R»
2 A(x) Y I -vx+Ax dv YI I { f v 110g v I e
o
AJX) vYle-vx+Av2dvl
~
o (1.22)
~
2II cos 2R
Y1
IA(x)
f
o
e
-
-vx+Av 2 Y I v 110g vldv ~
300 -
dv ~
AO
(1.22) -(y +1)
>
(HI + I)x
on choosing
I
suitably. From relations (1.19) to
H2
(1.22), we see that (b) 0 < x < 0(A)
(1.7) for (1.23)
<
0
! e -u
YI
> 0
y(x)
for
x
~
0(A).
: Applying (1.6) for
>
u
I
and
S I, we have
u
Y
log u -I+u Y I log u ! S B(u I log u) 2 •
Hence, for any fixed
>
a
0,
2
co
IRe f eiuX-AU {e- u
YI
Y
log u_l+u 1 10g u}du! S
o co _Au2 2Y I 2 S B feu (log u) du S
o 2 2y -a
I
(1.24)
I
-Au u J e
S B
co
du+
Setting
-(Y + I
I
'2
(I-a» +A
-(Y I +
I 2" (I +a»}
S
x
E; = 2A
Re
j
eiUX-AU2uYI
o
(1.25)
du S
I
0
S B{A
2 2y l +a
-Au u J e
2 x -4A
• Re e
co
J e
o Y
Re Ii)
+)
log u du •
-A(u-ix)2 Y I u log u du • 2
!!- E;
e -4A
J
e
A(v-0 2 Y) • 1t v (~2" +10g v)dv}
o -
30) -
+
x
+ Re {-i
For
A
2
-4A
yl+1
2
00
o
x S 0(A),
for all
Ilogl;l>][
",=AO'
Y
J e- Au (u+iO llog(u+iOdu}.
e
so that the
first term is dominated by 2 Y +1 -~ I; 2 R e { ~. 1 e 4A J e A(v-0 v Y 1 log v dv}
o
which is negative since (0,1;)
and
<
log v
0
in the interval
in view of the conditions on
Y 1 • The second
term is in absolute value
x
2
S Be- 4A
2
j
e- AU
Y
_ 1 -ex (u2+1;2) 2 du,
o etc. Assertion (ii) of Lemma 1.2 as well as the following assertions, which,
together with the two proved above,
complete the proof of Lemma 2.4.2 of [3J, are proved similarly. LEMMA 1.3. Let integer,
and
and
Then,
be real.
for all sufficiently large
(i)
be an even
2 YI exp(-At -It I logltl)
A
>
0, we have:
is a ch.f. i f
YI=O (mod 4); 2
Y1
(ii)
exp(-At -It I
(iii)
Y I =2(mod 4); 2 YI exp(-At -It I (a O a
ch.f.
log
2
Itl) is a ch. f.
y 1 =2(mod 4).
if
-
302 -
if
is
2. ON A BASIC LEMMA OF LINNIK'S IN THE CONTEXT OF FACTORIZING AN I.D. LAW WITH NORMAL COMPONENT - A CONJECTURE Denote by
the class of all infinitely divisible
IO
(i.d.) laws all of whose components are themselves i.d. In the course of establishing a necessary condition for an i.d.
law with normal component to belong to
I O'
Linnik enunciated three basic lemmas. The first of these three lemmas runs as follows: LEMMA. Let integers
(I
<
p
and
<
p
ciently small)
v
Then,
>
be relatively prime positive
and let
q)
positive constants.
q
0,
y,
for some
and
AI
(and so for all suffi-
the function
fv
given by
2 i t i t it fv(t)=exp[-yt +AI(e q -1)+A 2 (e p -I)-v(e -I)]
is a characteristic function.
An attempt to prove this by methods more elementary than given in the original proof of Linnik's ied to the following conjecture. We also give below a minimal discussion on it and indicate how a proof of the above lemma could be based on it. CONJECTURE. Let positive integers
(I
and
p
<
<
p
positive constants. Then, gv
q
be relatively prime
q) and
for some
y'YI' v
>
and
A2
be
0, the function
given by
has a power-series expansion in which the coefficients of
zn
are all positive for -
n ~ pq.
303 -
We make the following remarks: I. The conjecture is motivated by the following two
facts: (a) every integer form
where
ap+bq
~
n
~
and
integers (depending on
pq b
n):
can be written in the are suitable non-negative
cf. BIRKHOFF and MAC LANE
[I]
p. 20, ex.12. for
(b) in the case v-O, the expansion in power-series eXP(A l z q +A 2 ZP ) has, in view of fact (a) above,
strictly positive coefficients for all powers of greater than or equal to
Z
pq. The conjecture is thus that
this property will be preserved for small values of
v > O. 2. Some relations and estimates.
E c zn and E d zn denote respectively the n n n n power-series expansions for eXP(A l z q +A 2 ZP ) and q P eXP(A l z +A 2 Z -VZ), then we have Cauchy integral formulas for c and d which lead to n n (a) If
21!i(c -d )
n
n
=J
where
CR is the (standard parametrization of the) circle of radius R with the origin as centre. Setting
R= I,
for instance, we get a preliminary simple estimate
(b) The nC n ndn
c
=
n
qAlc
and
n-q
dn
obey the recurrence relations,
+PA 2 C n-p ,
qAld n-q +pA 2 d n-p -vd n- I. -
304 -
3. Once the conjecture is verified, we may proceed as follows:
if
f
were to be a ch.f. it would
\I
correspond to a distribution function which admits a continuous version
of its probability density func-
tion, given by
'"
f
21lp (x) \I
'"
const
~
n=O
const e
"'f
d
e
-itx-I/2t 2 itn d e t
n
1 2 - -x 2
2 n +nx
1
~
o
d
n
e
- 2"
By considering separately for a suitable (i)
p
\I
Ixl
S
>
(x)
xo
0
and (ii)
for all
that
(i) PO(x)
all
n
~
>
0
Ixl >
the sets
we can establish that
x O'
x, in view of the facts respectively for all
for suitable
pq,
xo
x \I
>
and (ii)
d
n
>
0
for
O.
REFERENCES [IJ
G.D. Birkhoff algebra,
[2J
S. Maclane, A survey of modern
3rd. ed., Macmillan,
1965.
B.V. Gnedenko - A.N. Kolmogorov, Limit distributions for sums of independent random variables, 2nd ed.
Addison-Wesley, [3J
1968.
A.M. Kagan - Yu.V. Linnik - C.R. Rao, Characterization problems in mathematical statistics, John
Wiley, [4J
1973.
Yu.V. Linnik, Decomposition of probability laws (in Russian), Leningrad, Izd. Leningr. Univ.,
[5J
Yu.V. Linnik -
1960.
I.V. Ostrovskii, Decomposition of
random variables and vectors Moscow, 1972.
-
305 -
(in Russian), Nauka,
[6]
B. Ramachandran, On the strong Markov property of the exponential laws, this volume, pp.
[7]
B. Ramachandran - C.R. Rao, Solutions of functional equations arising in some regression problems, and a characterization of the Cauchy law, Sankhya A., 32 (1970), 1-30.
[8]
R. Shimizu, Characteristic functions satisfying a functional equation 20(1968),
I, Ann. Inst. Statist. Math.,
187-209.
B. Ramachandran Indian Statistical Institute 7 SJSS Marg, New Delhi,
110029 India
- 306 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
ASYMPTOTIC EXPANSIONS IN A LOCAL LIMIT THEOREM* V.K. ROHATGI
1.
INTRODUCTION
Let
a sequence of independent, identically
{x } n
distributed random variables with common distribution function
and characteristic function
F
Elx 1 I m+ 2
<
00
for some
m
~
1
and that
f.
Suppose that 2
Ex 1 =0, Ex 1 = 1 •
Write
S
n
p (s
F (x) n
<
n
x
In) ,
S
it.....!!. f
n
(t)
=
Ee
In
Let the Cramer condition
*
Rest!arch supported by National Science Foundation Grant 77-01834. -
307 -
"ncs
lim
(C)
f(t)
I t I_co be satisfied.
<
I
It is then well known (see [4]. page 169)
that
F (x)
G(x)+o(n
n
-m/2
).
(I) ~(x)+~(x)
G(x)
m
~
n
-s/2
s= I
where
~
and
~
Q (x). s
are respectively the density and the
distribution function of the standard normal distribution and
Q (x)
is a polynomial coefficients of which are s defined by means of the semi-invariants A3 .A 4 ••..• Am+ 2 of the random variable XI. In [2] GALSTYAN has obtained conditions that ensure that m+o
-1+ -2-
(2)
SuplF (x)-G(x) n x
n
m=O
is satisfied. The case
I <
co
was investigated by HEYDE
in [3] where a necessary and sufficient condition for (I) was obtained.
Here we prove analogous theorems for densities. More precisely. suppose that for some iable
S
n
lin
(3)
PN(x). Then uniformly in
=
g(x)+o(n m
g(x)
the random var-
has an absolutely continuous distribution
with bounded density
P n (x)
n=N
~(x)+ ~
-m/2
).
n- s/2Q s(x)
s=1
-
308 -
x
as
where
(4 )
q
s
(x)
is the density of
and
S
n
lin
n ~ N.
for
See, for
example, Theorem 1.5 on page 206 of PETROV [4J. We will prove the following results. THEOREM I.
{x } n
Let
be a sequence of independent
identically distributed random variables with
Ex 2=1
Elx ' I
and
I
that for some
I
m +2
n=N
<
for some integer
00
the random variable
S
m
lin n
Exl=o, ~ I. Suppose has an
absolutely continuous distribution with bounded density If
Elx ,
I m+ 2 + 6
I'
<
if
00
0
<
6
<
I,
(5)
then
-1+ (6 )
m+6
-2-
n
sup I p (x)-g<x) n
x
I <
00
(0 :5 6
<
I)
is satisfied.
THEOREM 2. Let
{X}
be sequence of independent,
n
identically distributed symmetric random variables with 2 Exl=O, EXI=I.
for some for (7)
(6)
Suppose that
Snlln
has a bounded density
n=N. A necessary and sufficient condition
to hold is that
E{lx l l m +2 l0g(I+lx l l)} < -
309 -
00
if
m
is even for 6=0
otherwise.
(7)
The case
maO
was treated by GALSTYAN in [I] where
a necessary and sufficient condition for (6) was obtained. 2. PROOF OF THEOREM 1 The methods employed here are fairly standard. To avoid duplication we will follow GALSTYAN [2]. GO itx Let 1jI(t) - J e g(x)dx. Then -GO
where
P
s
is a polynomial in it of degree
3s, determ-
ined from the equation GO).
exp{ ~
s-3
s~
n
(it)s (s-2)!2} -
GO
1+ ~ P (it)n- s
/2
s= 1 s
Under the conditions of Theorem 1 we have
m+2 ('t)s f(t) - exp{ ~). ~ , + s-2 s s. where
aCt) - 0
as
t -
Itl m+
2
aCt)}
O. Hence
(8) where
m+2 a -
~ , (s-2)/2 • s-2 s.n
From Lemma 5 of [2] we see that (5) implies the existence of an A > 0 for which
-
310 -
A
(9)
<
la(t) I dt
f
o
t
1 +0
GO
•
It suffices to show that (9) implies (6). Clearly GO
(10)
~
f
Itl
n
f
If (t)ldt +
Itl>yln
n
I~(t) Idt
f
+
If (t)-~(t)ldt+
Itl>yln where Now
y=min(YI'Y2'Y3'~)
~
-1+ m+6
n
--2--
n~N
f
m+6 -1+2n
(t)-~(t)ldt ~
If
I t I ~y In
~
( 1 1)
as chosen on page 530 of [2].
f
n
Itl~yln
n~N
lea_~(t)ldt +
m+6 -1+ -2-
+
~
n
n~N
On applying inequality (6) of [2] we have TI ~ C ~ n
-1+ m+6
2
m+ 1
00
f n- -2-(ltI 3 (m+l) +
n~N
c f (ItI 3 (m+I)+
(12)
-
311
-
In case of
we use the inequality
T2
3t 2
- -8S e
It I S
which holds for that
aCt) - 0
as
t
Y21n
0
-
(Lemma 2 of [2]) and the fact implies that for some
Y3
the inequality t
t
m
lin I la(In)1
S 1/8
I t i S Y 31n.
holds for
so
Moreover
that m+6
-1+ -2~
T2 S
n
n~N
Y
J
S 2
o
u
m+2
2
( )1 1au
'<'
... n
(m+I+6)/2 e -nu /4d u.
n~N
By Abel's theorem 00
Ii. (1_5)1+6/2 ~ n 6 / 2 sn st I
r(l+
6 2).
n= I
It follows that (13)
T2 S
c
j
o
la(u) u l +6
I
du
Here and in the following
< "". C
is a generic constant.
It is clear from the expression for
(14)
~
n~N
n
-1+ m+o -2-
!~(t)ldt < "".
J It l>yln
-
312 -
~(t)
that
>
0,
Since
is bounded,
PN(x)
Consequently
max
If(t) I
=
I f(t) IN -
e- C
<
0
as
It I -
co.
1. Moreover
Itl~y
I
!f(t) N EL 2 (-co,co).
Hence
m+o -1+ -2l: n
f
m+o -1+ -2(15)
I f (t) Idt
Itl~yrn
I f(..!...) I
dt S
I f(fn) I
In
Itl>yrn
n~N
n-2N
2N
f
n
l:
n
m-I+o l:
S
2
n
e
-(n-2N)c
n~N
f
If(t) 1 2Ndt
<
co.
Itl>y
In view of (II) through (15) the proof of Theorem is complete. 3. PROOF OF THEOREM 2 In view of what has been shown above we need to show that for symmetric random variables (6) implies (9). For
n
~
N
f
n
we have (t)-'¥(t)
f
e
itx
(p (x)-g(x»dx. n
Also, co
I2i(f
e
itx e -x 2 /2d x.
-co
From Parseval's identity we obtain co
f
(f (t)-'¥(t»e-
t2/2
n
dt =
co
I2it f
-co
In view of (6), we have -
313 -
e
_t 2 /2
(p (t)-g(t»dt.
n
m+6
- I + -2-
n
l:
00
II
n~N
e- t
2/2
(f (t)-~(t»dtl s n
_00
-1+
m+6
--2-- sup l p (t)-g(t) Idt t n
<
00.
Under the conditions of the theorem the functions ~
and
a
are real and even, so that -1+
m+6 --
2
l: n
""
II
o
n~N
_1+ m+6
l: n n~N
2
I
yl'ii 2 I [f (t)-~(t)Je-t /2 dt 0
I [f (t)-~(t)J yl'ii n
+
n
2
00
+
fn'
e- t
/2 < "".
This is equation (19) of [2]. Repeating the argument in [2] one shows easily that (9) holds. The proof of Theorem 2 is now complete in view of Lemma 7 of [2]. REFERENCES [I]
F.N. Galstyan, Local analogs of a theorem of Heyde, Soviet Math.
[2]
Dokl.,
12(1971),596-600.
F.N. Galstyan, On asymptotic expansions in the central limit theorem, Theory of Probab. and its Appl.,
[3]
16(1971),528-533.
C.C. Heyde, On the influence of moments on the rate of convergence to the normal distribution, Z. Wahrscheinlichkeitstheorie verw.
[4]
Geb., 8(1967),
12-18.
V.V. Petrov, Sums of independent random variables, Springer-Verlag, New York, 1975. -
314 -
V.K. Rohatgi Dept. Math., Bowling Green State Uni Bowling Green, OR 43403, USA
-
315 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION UETHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY), 1977.
CHARACTERIZATION AND TESTING FOR NORMALITY K. SARKADI
This paper deals with relations between characterization problems and testing for normality.
XI' ... 'X n
Throughout the paper, let
denote
independent, identically distributed random variables
F(x). Further, let
with distribution function
FO(X) be
a given probability distribution function with (I)
0,
(2 )
I.
Let
F
tions
F(x)
eters
(0
by
>
HO.
Let
X
denote the parametric family of distribuF (x-~) where ~ and 0 are the paramO 0 0). The hypothesis F(x)EF will be denoted
=
n I: X.,
n i-I
~
5
=
[
~
i=1
- 317 -
(x._X)2] 1/2 ~
TRANSFORMATION METHODS
I.
ThroughoQt this section let
~(x)
FO(X)
be the
standard normal distribution function. Some methods for testing normality ([6J, pp. 49-54, [12J, [2)]) consist
of two steps. The first step is a
parameterfree transformation. The resulting variables are identically distributed and independent, provided HO
is true. Their distribution under
HO
is completely
specified (or depends on at most one nuisance parameter). In the second step, a
(simple)
goodness of fit test is
performed on the resulting variables. For example, the transformed variables n-I ~
(3)
x.
x.-(j=1
Y.
~
~
X
] + -.E..)
(i=I, ... ,n-l)
In
n+1n
are independent and have the distribution the distribution of the variables
x.
~
is
2
N(O,a ) N(ll,a
2
if
)
([IIJ, p. 270). Conversely, the normality of the variables
implies that of the initial variables, as
Y.
~
this follows from Cramer's theorem ([3J, p.53). Some transformations (27); [2IJ, Eq.
(4) where
Y.
~
U
variables.
=
and
(10»
([12J, Eq.
(7.1); [6J, Eq.
can be written in the form
(X.-u)/v ~
V
are again functions of the initial
In these transformations, the variables
have the distribution
N(O,I)
Y.
~
is true. The
converse statement is not as generally valid as in the former simple case. However, the following theorem holds:
-
318 -
THEOREM I. Let XI' .•• 'X k (k ~ 6) be independent identically distributed random variables and U and V arbitrary random variables.
for all combinations ••• ,6
(i,j,k,~)
without repetition.
i=I,2, •.. ,6)
Suppose that*
of the elements I ,2 •••• Xi-U If the random vector ( --V- ,
is normally distributed this implies the
normality of the initial variables
XI' ... ,Xk .
The theorem follows easily from a known characterizing property of the normal distribution with expectation zero, see e.g.
[I],
Theorem 9.2. More detailed proof is
given in [14]. The transformation methods of testing for normality when first proposed ( [ I I ] ; [6], Eq.
(26); [21]) meant
the first possibilities to perform an exact test for normality. But even now they are important from the point of view of the application, in particular, in the multisample case.
(The artificial sampling evaluation in [4]
considers the one-sample case only.) We call multisample normality test such a test in which we assess the normality of a number of samples. The underlying populations may differ in their moments. In practice, it is often hard or impossible to provide a sufficiently large series of observations for assessing normality but it is possible to take a sufficient number of observations in short series or past data of this kind are available.
*
The notation
d =
means the equality in distribution.
-
319 -
Even if the expectations of the parent populations differ but the variance can be supposed to be constant (homoscedastic case), the transformation (3) can be applied within each sample. The second step (a test for normality in the traditional sense) is applied for the unified series of the transformed variables. In the heteroscedastic case, a transformation of type (4) can be applied in the first step. Appropriate tables [10] are helpful in the computation work. Practical interest suggests further investigation of the characterization problem for small sample sizes. 2. PROBLEM OF EXTENSION We may ask whether method given in the last Section can be extended for the case when
FO(X)
is not normal.
The answer is that, under fairly general conditions, it is possible to provide independent and identically distributed transforms [9]. The functional relationship is, however, more complicated than in the normal case and computational difficulties make the application practically unfeasible in the general case. For other special cases (exponential and Gamma distributions) we refer to [15]
and [12].
One of the basic properties of the normal distribution which makes a relatively simple
generation of
independent transforms possible is that independent linear transforms of the sample elements exist. According to the theorem of SKITOVICH and DARMOIS ([8], pp. 89-90) this property characterizes the normal distribution.
-
320 -
3. SHAPIRO-WILK TEST In this Section, let us suppose that
FO(X)
is
strictly monotone and continuously differentiable up to the third order on the whole real line. Let us suppose 2 further, that lim x fO(x) = 0 where fO(x) - FO(X)' x ..... ±oo Let X ln S ••• S Xnn denote the rearrangement of the series
XI""'X n
according to the order of mag-
nitude. Let us denote
E(x.
m.
~n
~n
!F(X)
= FO(X»
and determine the constants
a.
in such a way that
~n
n ~
i= I
a.
~n
=0,
n ~
i=1
a. m. ~n
~n
and
min. n
For
it follows that
F(x)EF
~
i=1
variance unbiased estimator of
a
a. X. ~n
~n
is the minimum
among the linear
combinations of the order statistics. The test statistic of the Shapiro-Wilk test [17] is n
W
....!....2( S
~
i= I
a. X. ~n
~n
)2
(here and in the sequel unimportant constant factors are neglected). and that of the Shapiro-Francia test [16] is
-
321
-
n
W'= _1_ ( ~ 52
m. X. ~n
i=1
~n
)
2
•
These two tests have been put forward for testing
HO.
the validity of the hypothesis
([2J, p.68) proved
CHERNOFF, GASTWIRTH and JOHNS that T
n
is an asymptotically efficient estimate for the scaie
L'[F~I(u)1
J(U)
(5)
F(x)EF. Here
a, if
parameter
where L(x)
(6 )
II
and
I2
function
are constants depending on the distribution
FO(X)
(II,I 2 )' is the second column of the
inverse of the matrix (3.5') in [2J, p. 67. If
fO(x)
is symmetric:
fO(x)
=
fO(-x)
then
II=O,
fO(x) XLi(X)fO(X)dX]-1 i n+1
a:
Let
where
-x
f(x)
n
-
J(---)-J
~n
L 2 (x)
where
J
~
n i=1
.
~I)
J(
is an
n
asymptotically vanishing correction. It assures that n will be parameterfree in respect to the loca~ X. i=1
a:~n
~n
tion parameter
ll.
n
a:
can be considered the asymptotic ~ X. n i= I ~n ~n n a.~n X.~n and the test based version of the statistic ~ Then
i=1
on the statistic -
322 -
n ...!....2( ~ 5 i= I
V
a~ x. ~n
~n
)2
is the asymptotic version of the Shapiro-Wilk test. The asymptotic version of the Shapiro-Francia test is the deWet-Venter test [5] with the test statistic n
_I ( ~
v,
52 i=1
where
(We remark that [5] defines the test for the case of normal null hypothesis. In that case the second term m~
in the expression of
~n
In the case when
vanishes.)
FO(X)
(cf. [2], p. 70) and thus
is normal
J(u)
=
-I FO (u)
v=v', i.e. the asymptotic
versions of the Shapiro-Wilk test and Shapiro-Francia test coincide. Accordingly it has been known empirically that and
a in
are nearly proportional in the normal case
m.
~n
([17], p. 596, see also [20]). In addition, TusnAdy has proved [22] that lim (a[ if
0
<
y
ny
]
,n
-I FO (y)
In)
<
is normal
(the proof is
cumbersome). Sampling studies revealed that the tests of normality of Shapiro, Francia and Wilk [16], [17]
are
superior to most of the classical tests in the sense that they have good power against practically important alternatives [7], [16], [19]. -
323 -
The Shapiro-Francia test has been proved to be consistent in the case of arbitrary
HO [13J.
On the other hand, the Shapiro-Wi1k test may loss its good properties if
differs from normal CI3J,
FO(X)
CI8J. The above circumstances raise the question: is there any relation between the merits of the ShapiroWi1k normality test and the asymptotic equivalence of Shapiro-Wi1k and Shapiro-Francia tests in the normal case? Also, the following related question may be of interest: Apart from the normal case, is there any other form of
FO(X)
for which the tests based on
V
and
V'
are equivalent? THEOREM 2. Under the assumptions of this Section, the following statements are equivalent: a) For a given significance level
sample size
a
and for every
n, the asymptotic versions of the Shapiro-
wilk and Shapiro-Francia tests coincide. fO(x)
b)
has the shape
2k+1
(~)--2-(7)
x
f(2k+l)
2
-(2k+I)~
2k e
2
where
k
is a nonnegative integer.
PROOF. Owing to the assumptions, the joint density function of Xln' ••• 'X nn under the null hypothesis is positive almost everywhere in the interior of the domain xI S x 2 S ••• S x n • V and v, are continuous functions of its variables Xln' ••• 'X nn excepted for the discontinuity line x l =x 2 =••• =x n •
-
324 -
Let and
V',
v and v' denote the critical values of V n n respectively. If a) is true the boundaries of
the critical regions must coincide, i.e. the equations and V'=V ' n n transformation
are equivalent. Performing the
V=v
(8)
x .-x s ~
Y.
~
(i=1,2, .•. ,n) n
we obtain the equations
~
i=1
n
a~Y.) ~n
~
2
=v
n
and
Y.) 2 = v' , respectively. However, transformam~ ~ ~ ~n n i=1 tion (8) maps the n-dimensional Euclidean space onto (
the set determined by the equations n ~
i=1 n
Y.
0,
~
2
Y.
~
i=1
I ,
~
therefore the equation systems n
Y. I 1. i-1 n
I i-I
Y~~
-
0,
-
I •
n I Yi I I a ~n i-I
I
-
IVn
and
Y.
-
Y~
'" 1 ,
n ~
i"'l n I i"'l
~
~
0,
- 325 -
n
.r
Y.I ~
m'
in
i=1
are equivalent.
IV' n
=
Simple calculation shows that this impli-
es
~
Im ~
~ I =
a ~n - a In
~n
~
-m In
I
(i.j=1 ,2, . . . ,n).
IV' n
IV n'
Taking the definitions of
and
m~
~n
into
account we obtain that i
j n+1
J(-)-J(-)
(9)
n+1
(i,j=1
where
c
2 n
,2, ...
v Iv'. n
n
It follows from which means that
c
(9)
n
that
n+1 !N+I
and
c
n
=c
N
n. T
n
and from
(2) follows 1
J J(u)du
o )
1
_)
J FO (u)du
-)
o
consequently
0,
o
J J(U)F O (u)du
and
implies
is independent of
From the asymptotic efficiency of (I)
,n)
1
_I
2
J [FO (u)] du
1,
o
-I
c=I,J(u) n
F0
and, by virtue of
(u)
(6), x
L(x)
2
'"2
By integration we obtain
-
326 -
+ c.
(5)
In order to be this expression a continuous density function it is necessary that teger and
I2
>
~
o.
be a nonnegative
II ~ 0 and x > 0 imply II ~ 0 implies _!xfO(x)dx ~ 00
therefore
II=O.
Taking (I) into account we obtain The constants
-L
Co
00
and
=-L
00
2
I2
are determined by the
fO(x)dx x fO(x)dx thus it is a necessary condition.
conditions
=
I.
We obtain (7)
Straightforward calculation shows that
J(u)
in-
O.
It is easy to see that
fO(x) ~ fO(-x),
k
-I
F0
implies
(7)
(u)
and thus, at the same time the condition is sufficient.
k=O
yields the normal distribution. This seems to be
the only case of practical interest. In case
>
k
0
the
distribution is bimodal.
REFERENCES [IJ
L.
Bondesson, Characterizations of probability laws
through constant regression, theorie verw. Geb., [2J
H.
z.
Wahrscheinlichkeits-
30(1974), 93-115.
Chernoff - J.L. Gastwirth - M.V. Johns, As-
ymptotic distribution of linear combinations of functions of order statistics, Ann. Math.
Statist.,
38(1967), 52-72. [3J
H. Cramer, Random variables and probability distributions,2nd ed.,University Press, Cambridge,
1962. - 327 -
[4J
T. Deutler - H. Griesenbrock - H. Schwensfeier, Der Kolmogorov-Smirnov-Einstichprobentest auf Normalitat, Allgem. Statist. Archiv., 59(1975), 228-250.
[5J
T. DeWet - J.H. Venter, Asymptotic distribution of certain test criteria for normality, South. Afr. Statist.
[6J
J.,
6(1972),
135-149.
J. Durbin, Some methods of constructing exact tests, Biometrika, 48(1961), 41-55.
[7J
A.R. Dyer, Comparison of tests of normality with a cautionary note, Biometrika, 61 (1974),
[8J
185-189.
A.M. Kagan - Yu.V. Linnik - C.R. Rao, Characterization problems in mathematical statistics, Wiley,
New York, [9J
1973.
F.J. O'Reilly - C.P. Quesenberry, The conditional probability integral transformation and applications to obtain composite chi-square goodness-of-fit tests, Ann. Statist.,
[10J L.
Sallay - K.
1(1973), 74-83.
Sarkadi, Test of normality based on
a number of small samples, 20th Conference of European Organization of Quality Control, Copenhagen,
1976, 21-27. [I IJ K.
Sarkadi, On testing for normality, Publ. Math.
Inst.
[12J K.
Hung. Acad. Sci., 5(1960), 269-274.
Sarkadi, On testing for normality, Proc. 5th
Berkeley Symp.
on Math.
Statist. and Prob.,
1(1967),
373-387. [13J K. Sarkadi, The consistency of the Shapiro-Francia test, Biometrika, 62(1975), 445-450.
-
328 -
[14J K.
Sarkadi, Testing for normality, Publ. Banach
Institute,
[15J K.
6(1979), to appear.
Sarkadi - G. Tusn!dy, Testing for normality and
for the exponential distribution, Proc. 5th Conf. Probability Theory, Brasov, Acad. R.S.R. Bucharest
1977, 99-118. [16J S.S.
Shapiro - R.S. Francia, An approximate analysis
of variance test for normality, J. Amer. Statist. Assoc., 67(1972), 215-216.
[17J S.S.
Shapiro - M.B. Wilk, An analysis of variance
test for normality, Biometrika, 42(1965), 591-611. [18J S.S. Shapiro - M.B. Wilk, An analysis of variance test for the exponential distribution, Technometrics, 14(1972), 355-370. [19J S.S. Shapiro - M.B. Wilk - H.J. Chen, A comparative study of various tests for normality, J. Amer. Statist. Assoc., 63(1968),
[20J M.A.
1343-1372.
Stephens, Asymptotic properties for covariance
matrices of order statistics, Biometrika, 62(1975), 23-28. [21J H.
Stormer, Ein Test zum Erkennen von Normalver-
teilungen, Z. wahrscheinlichkeitstheorie verw. Geb., 2(1964), 420-428. [22J G. Tusn!dy, personal communication (1974). K.
Sarkadi
Mathematical Institute of the Hungarian Academy of Sciences 1053 Budapest Re!ltanoda u.13-15 Hungary - 329 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION IffiTHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
CANONICAL CORRELATIONS, RANK ADDITIVITY AND CHARACTERIZATIONS OF MULTIVARIATE NORMALITY* V.
SESHADRI - G.P.H.
STYAN
1. INTRODUCTION AND SUMMARY OF RESULTS SESHADRI and PATIL [8J considered characterizations of some bivariate distributions by the marginal and the conditional distributions of the same component, while BHATTACHARYYA [2J was among the first to study sets of sufficient conditions leading to bivariate normality. The general problem of determining the bivariate distribution via the marginal distribution of a component and the conditional distribution of the same component is related to the problem of mixtures and their identifiability, first investigated by TEICHER [9J, [10J. In this paper two characterizations are obtained for the random vector al
~ = (~~)
to be multivariate norm-
(mvn). Let the random vector
~
have
cova~iance
* Research supported in part by the National Research Council of Canada and by the Gouvernement du Quebec, Programme de formation de chercheurs et d'action concertee. -
331
-
matrix
E
=
(~11~12)
, possibly singular. It is shown
21 22
that the number of unit canonical correlations between ~I
~2
r(E II )+r(E 22 )-r(E), where r(·) denotes rank, extending a result of KHATRI [5]. This leads to the and
is
~
following characterizations for
to be mvn when the
~I 1~2
conditional distribution of
is
r(E),
(a)
and
(b)
~I
N(PI~2,QI):
and
is
x Ix -2 '-I
is
2. CANONICAL CORRELATIONS AND RANK ADDITIVITY Consider the
(I)
where
x
random vector
pxl
= ( ~~) ,
~I
is
~2
and
PIX)
is
P2 XI • Then HOTELLING
[4J found the linear combination
(2)
JL-
If"
say, where (a)
is
pxp, so that
the components of among
(b)
M
. (~~),
themselves~
the betw~en
:£.1 and
and of
:£.2
are uncorrelated
cross-correlation matrix, R, say, !L)
and
!L2
has all off-diagonal elements
zero and all on-diagonal elements, PI say, as large as possible: -
332 -
~ ••• ~
Ps
> 0,
(3)
R
The ~I
p.'s
0
0
o
P 2 ••• 0
0
0
o o
0
o
0
o o
0
o
~2'
and
~
the vector of canonical variates.
As will be shown below,
PlxP2
o
0
are called the canonical correlations between
~
and
PI 0 ••• 0
s
equals the rank of the
cross-covariance matrix,
~I
E 12 , between
and
~2'
The statistical aspects of canonical correlations are presented in some detail by ANDERSON [IJ 12) while
(Chapter
BJORCK and GOLUB [3J give numerical methods
for computing them. Let us write the covariance matrix of
x
as
(4)
where that
(5)
where
EI2
'
PI
is
... ,
P l xP 2 · ANDERSON [ I J , p. 290 has shown are roots of the matrix equation Ps
( - pEl I E I 2 ) E21 -pE 22 ;: ;:
0
is non-null, pxl. Let
and full rank decomposition
-
333 -
E
have rank
r S p,
(6)
where
XI
is
and
X~X .
I: ..
(7)
rXPI
~
~]
X
is
2
rxP 2 • Then
(i,j=I,2).
]
We now write singular value decompositions of and
X2 ,
r.
~
= r(X.) = ~
r(I: .. ) ~~
XI and
U.D.V~
X.
(8)
i=I,2,
so that for
~
~
~
~
where U~U.
(9)
and
~
D.
~
V~V. ~
I
~
r.
~
is diagonal positive definite
~
r.Xr .. Then the ~
~
matrix in (5) becomes
(10)
where
Thus
(5) becomes
(12)
Since
B
B
(
-pI
rl U'U 2 I
UjU 2 ) -pI
r 2
B'u
= O.
--
has full column rank it follows that (12) has
a non-trivial solution
B'u
-
if and only if 334 -
I-pI
UjU 2
II
(13)
-pI
UiUI and so
PI
'
... ,
0,
I2 are the positive roots of
Ps
( 14)
0,
or the nonzero singular values of
UjU 2 , cf. BJ5RCK and
GOLUB [3J. Hence ( 15)
as claimed below (3). We now prove that (16)
Pi S
I,
and that the number equal to
is
Replacing ( 18)
where
BeB'
B
n, say, of canonical correlations
P
by
-I
in (10) yields
,
is as in (II) and
( 1 9)
The Schur complement
-
335 -
(elI
(20)
r 2
)
=
I
r l
-U'U U'U
I 2 2 I
is nonnegative definite since
C
is and so (16) follows
at once. To prove (17) we note that rank is additive on the Schur complement (MARSAGLIA and STYAN [6], p.291), so that
r
(21)
=
r(E)
=
r(C)
=
r 2 +r(C/I
r
2
);
the number of unit canonical correlations, however, is the nullity of (20), i.e., n
(22)
=
r
I
-r(I
r l
-U'U u'u ) I 2 2 I
=
r
I
-r(CII
r
) 2 '
cf. KHATRI [5], p. 469. Adding (21) and (22) yields (17) Using (6) we may write (17) as
C(·)
where
denotes column space. Since the matrix
X
in (6) has full column rank it also follows that
(24)
When
n
E
= is positive definite, of course,
n=O.
3. CHARACTERIZATIONS OF MULTIVARIATE NORMALITY Suppose now that the random vector as in (I), has the covariance matrix in (4). If in addition
-
336 -
~,partitioned
E, partitioned as
where the matrices nonrandom,
PI
(P I XP 2 )
and
0 1 (PIXP I )
are
then
(26)
( I PI)
~2 ;
P2
Hence
P L Pj+OI [ I 22 L 22 Pj
Therefore
using (6). ~
Since
(18)
is consistent and if
LI2
then
0
(29)
for some p.
g-inverse
c f.
L22
e. g.
RAO [7 ] ,
24. Suppose further that in addition to
have
- 337 -
(25)
we also
P 2 (P2 XP I) and 02 (P 2 XP 2 ) are nonrandom. Then in parallel with (27) we obtain
where the matrices
E .. (
(31)
so that, cf.
P
(32) for some
2
EIIPi Ell P 2 E II P2EIIPi+02) (29) , i f
= E2IE~1 g-inverse
..
E21 = EI2 XiX
(
Ell EI2 ) E21 E22
'¢
0,
I (x j X I)
(XjX I )-.
Ell
We now assume that both
~I 1~2
and
~21~1
are
multivariate normal, i.e.,
(33)
Then the moment generating function (mgf) of
(34)
Similarly, the mgf of
~2
is
(35 )
Thus (36)
-
338 -
~I
is
Let
Then from
(31),
(38 )
using
(27).
Substituting
(38)
into
(36)
yields
and so
(40)
Substituting
(40)
into
(39)
yields
(41) (k=I,2, ••• ).
It
follows
(42)
then that
Ak -
0
as
if k
_
00
-
339 -
then (41) reduces to the mgf of (37),
N(Q,L 1 I ) ' Now, from
(29) and (32)
has the same nonzero characteristic roots as (44) using (8), and hence the same nonzero roots as
UjUlUiUI'
From (14) we then see that these nonzero roots are the squares of the canonical correlations between
~I
~2'
and
the roots of
PI,P2""
'P s
'
Now (42) holds if and only if all
are less than one in absolute value.
A
Thus it follows at once from (17) that
(42) holds if
and only if (45)
r (L) •
~I
In that event
Eexp!'~
~
N(Q,L 1 I ) ' and so
= Ex E{exp(!j~I+!i~2) I~I} -I
Eexp
{(!i+!iP2)~1
+ t ! i Q 2!2}
exp t{!jL 1 1!1+2!jLIIP2!2+~i(P2LI I P i+ Q2)!2} t
exp
t(!j'!i)L(~I)
1
exp "2 !'r!.
-2
We have, therefore, proved the following theorem:
-
340 -
THEOREM I. Let the random vector covariance matrix
(b)
~I
Ell E = [E 21
and
E12) E ' 22
x.
poss~bly
[ ~~21)
have
singular.
If
have no unit canonical co-
rrelations, (c)
where
(Uj;
PI'
P2 ,
01' and
i,j=I,2)
are nonrandom matrices then
02
x~N(Q,E).
Now we suppose that condition (c) in Theorem I holds and that
Then I
,""
exp "2 !I '"II!I
(48 )
and so Eexp!.iPI~2
I
exp
"2 !.j ( EII-O I )!.I
exp
2" !.jP I E 22 Pj!.1
(49) I
using (27). Hence
-
341
-
If
has full column rank then it has a left-
PI
inverse and in that event clearly
More generally suppose that theorem of ANDERSON [IJ,
E~2
=
~2.Then
using a
(pp. 25-26) we may write
, and
where Hence, using (50), we get
and so using (50),
(52) and (53) we obtain
(54) If
then we may cancel
PI
in (54), cf. MARSAGLIA and STYAN
[6J, p. 271, and so
which, using (52) again, yields (57)
x -2
~
The rank condition (55) simplifies to - 342 -
r(P l v 2 )
since
(28) . Normality of the
vector
pxl
~
then follows from
(51) or (57) using an argument similar to (46). We have, therefore, also proved THEOREM 2. Let the
I:
covariance matrix
=
If
then
=
random [ I:I: II 21
(a)
r(I:12)
(b)
~I 1~2 ~ N(PI~2,QI)'
(c)
~I?' N(Q,I:II)'
x
~ N(~,I:),
vector
~
=
[~I]
ha ve
I: 12J -2 I: ,possibly singular. 22
r(I: 22 ),
where
~
O.
REFERENCES [IJ
T.W. Anderson, An introduction to multivariate statistical analysis, Wiley, New York,
[2J
A.
1958.
BhattachanYla, On some sets of sufficient
conditions leading to the normal bivariate distribution, Sankhya, 6(1944), 399-406. [3J
A.
Bjorck -
G.H. Golub,
Numerical methods for
computing angles between linear subspaces, Math. Comp.,
[4J
27(1973), 579-594.
H. Hotelling, Relations between two sets of variates, Biometrika, 28(1936), 321-377.
-
343 -
[5]
C.G.
Khatri, A note on multiple and canonical
correlation for a singular covariance matrix, Psychometrika, 41 (1976), 465-470.
[6]
G. Harsaglia - G.P.H.
Styan, Equalities and inequal-
ities for ranks of matrices, Linear and Multilinear Algebra, 2(1974), [7]
C.R. Rao, Linear statistical inference and its applications,
[8]
269-292.
V.
Second Ed., Wiley, New York,
Seshadri - G.P.
1973.
Patil, A characterization of a
bivariate distribution by the marginal and the conditional distributions of the same component, Ann.
[9]
Inst.
Statist.
Math.
Tokyo,
15(1964), 215-221.
H. Teicher, On the mixture of distributions, Ann. Math.
Statist.,
31 (1960), 55-73.
[10] H. Teicher, Identifiability of mixtures, Ann. Math. Statist.,
32(1961), 244-248.
Note added in proof. After presentation of this
paper the authors have become aware of related results in [IIJ
C.G. Khatri, A characterization property of a normal distribution, Gujarat Statist. Rev., 2(1975), 24-27.
V. Seshadri and George P.H. Styan Dept. of llathematics, HcGi11 University Burnside Hall, 805 Sherbrooke Street West Montrial, Quibec, Canada H3A 2K6
-
344 -
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION HETHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY). 1977.
INFINITE DIVISIBILITY OF MIXTURES OF GAMMA DISTRIBUTIONS F. W.
STEUTEL
1. INTRODUCTION AND PRELIMINARIES In this paper we consider probability distributions of nonnegative random
variable~
(r.v.
IS)
with Laplace
transforms of the following type: ( 1. 1 )
with
~
ak
O. and its generalization, obtainable as a
limit of Laplace transforms of type (1.1), 00
( 1 .2)
1
a(x)
J (l+e) o
dG(x).
We are interested in conditions on the
Pk
and
ak
that ensure the infinite divisibility (inf div) of (1.1), and similarly for
(1.2).
Most of our results can be found in Chapter 2 of [6],
(only part of which has been properly published;
see [7] for references), to which we refer for complete proofs. One of the reasons for presenting these results, -
345 -
which hopefully have some interest in themselves, here, is to provide a background for two conjectures - one in probability and one in analysis -
that I have been trying
to prove for some time, whithout too much success.
In the
last section we briefly discuss some related results, quite recently obtained by others. We shall use the following notation.
If
X
a r.v.,
then we denote its distribution function
by
or
its probability density function
F,
~
0
is
(d.f.) (p.d.f.) ~
or
by
F,
its Laplace transform (L.T.) by
f,
and its characteristic function
i.e. we have, writing
E
(c.f.)
or
FX
or
by
(j) •
for expectation
We shall need the following well-known results
(cf.
[2]) .
LEMMA I. I. F
is inf div i f f 00
exp{J
o
where
F
KF
(e
-TX
-I)x
-I
F
has the form
dKF(X)}
is nondecreasing.
COROLLARY 1.2. If, in the notation of Lemma 1.1, n II F., then J
n ~ j=1
KF .
J
~
LEMMA 1.3. If
{A/(A+T)}a,
F(T)
absolutely continuous with
-
K;'(x)=ae
346 -
-AX
then •
is
LEMMA I .4. F
lim
F
n- oo
If
F
n L. T.
is the
n
If
LEMMA 1. 5.
is inf div for
~
of is
and i f
nEN,
d.f.,
a
then
is inf div.
F
inf div c. f. ,
an
then
[ 10] ,
p. 26 I )
tER.
for
From Lemma 1.4 and the fact that
(cf.
( I .3)
with
c(a,B)
>
it follows that it is sufficient to
0,
prove the inf div of
Pk
we take all
>
O.
(1.1)
for integer values of
If we allow the
Pk
ak ,
if
to be negative,
then we can restrict attention to the case
a=1
only,
as we have
which is essentially a linear combination of functions
A/(A+T).
of the form
2.
THE CASE
a=1
First we take all
Pk
>
and we suppose without restriction that
<
ak=l,
in (I. I) and all
0
<
0
Al
<
Then for the corresponding L.T. we have
A • n
n
n
Ak
n-I
n Ak +TI j=1 k=1
(2. I )
where the
are the
-llj
zeros of
n-I
F.
_llJ' 11
•
t-'J
+T '
From Corollary
1.2 and Lemma 1.3 it follows that n-I
n
(2. 2)
K' (x) F
=
~
~
k=1
j=1
e
One easily verifies, by considering the changes in sign
-
347 -
of
F
at its poles
A. ]
(2.3)
<
<
ll. ]
It follows that
F
-AI, .•• ,-A n , that one has
A.
]+
(j=1,2, ••. ,n-I).
I
in (2.2) is positive, and hence that
K' F
in (1.2) is inf dive Using (1.3) and Lemma 1.4 we then
obtain
o
~
THEOREM 2.1.
If
for
0
~
a(x)
I
~
(2.4)
J (_1_) o I +TX
is a d.f.
G ~
~
x
~,
on
[O,~J,
and i f
then
a (x) dG(x)
is an inf div L.T.
COROLLARY 2.2. ~
(2.5) is an
J
cp(t)
o
If
is a d.f.
G
I a. (I-it) dG(a.)
inf div c.f.
In [8J, D.
REMARK.
szAsz,
a proof of the inf div of [O,~J.
on
on [O,IJ, then
at 3, then
However, if cp
cp
wrongly attributes to me in (2.5) for arbitrary
has mass 1/9 at
G
0
G
and 8/9
is easily seen not inf div by Lemma 1.5.
The example he has in mind is provided by replacing 1.'n (2.5) by a c .f . (I -;t)-I •
th a t
. 1.S even an d 1 ogconvex
(O,~)·~.
on
For L.T. 's of the form (2. I) and its obvious generalization we have the following representation
*
See also S.J. WOLFE, Mixtures of infinitely divisible distribution functions, presented at this colloquium.
-
348 -
theorem, which, for this case, replaces Lemma 1.1 •
...
THEOREM 2.3. F
(2.6)
J o
with
a d.f.
G
on
is a L.T. of the form
- dG(x}, I+,x [O,~}
F(,}
exp{-J
o
m
where $
F
has the form
,
...
(2.7)
m(B}
iff
I..(I..+,}
dm(l..}},
is an absolutely continuous measure satisfying
I..(B}
for every Borel set
B,
A
with
denoting
Lebesgue measure.
PROOF (sketch). From Lemma 1.1 together with (2.2) and
~
(2.3) we obtain, putting
n
exp{J -e__-_'_x_-~I
o
x
=~
n
~
'
(e
-I.. k x
-e
-~kx
)dx}
k=1
exp{-J I..(~+'} m'(I..)dl..},
o
where (2.8 )
m'O)
=l~
if
< A <
A. ]
~.
(j=1,2, ... ,n)
]
otherwise.
It is now a technical matter to prove L.T. 's of the form (2.6).
Similarly, proving (2.6) from
(2.7) is a matter of approximating special
(2.7) for general
m
in (2.7) by
m's satisfying (2.8) and taking limits. For
details we refer to [6J. From Theorem 2.3 we take the following
-
349 -
COROLLARY 2.4. I f then
Gn
there eX.ist such
G
G
and
are
uniquely determined
d.f.'s on
d.f.'s
G
[0,""), and
CI.
that
{I
o
I+TX dG(x)}CI.
""
= I
(0
I+TX dGCI.(x)
~
I lIn n (I+TX) dG(x)}
""
{I
o
CI.
~
I),
0
(nEN) •
We note that from Theorem 2. I with
CI.(x)=1
we
obtain COROLLARY 2.5. Completely monote p.d.f.'s on
(0,"")
are inf di v. y
PROOF. F(T)
I
o
A A+T dGO)
iff
= ""I o
f(x)
Ae-AXdG(A).
REMARK. This result provides simple counterexamples to the following conjecture of A. RtNYI: X
and
mean zero.
is completely monotone for all
otone, then ~
8+1
8
= o,
~
By choosing
I .
or
made inf
If the r.v. 's
are both inf div, then X is normal with If fx(x) = x 8 g(x), with g completely mon-
x2
g(x)
=
8
const·e
div. A similar
or -x
g
)
p
~
in a suitable way (e. g. the p.d.f.
(unpublished?)
by D.G. KENDALL and J.F.C. KINGMAN.
fx
can be
remark was made
lowe this Remark to
E. LUKACS. Admitting negative
Pk
in (2.1) leads to diff-
iculties. We have the following result, which generalizes Theorem 2.1. THEOREM 2.6. I f in the sequence
(2.1),
PI,P2, ••• ,Pn
-
with
Al < A2 < ... < An'
has at most one change in
350 -
sign ,
...
then
is inf div.
F
For more than one change in sign however, there seem to be no general results as is shown by the following examples.
~(t)=2(I-it)-1-6(3-it)-I+
EXAMPLE I. The c.f. +S(S-it)
-I
is not inf div as
~
~(.IS)=O
(cf. Lemma I.S) .
...
EXAMPLE 2. F(T) n
we have
F(T)
=n k=1
REMARK. F
in
••• ,X ), where the n
is inf divas k
k+T
Ex~mple
2 is the d.f. of
max(x l , ...
X's are i.i.d. and exponentially
distributed. Rather surprisingly, in the discrete analogue: NI, ••• ,Nn i.i.d. and geometrically distributed, max(NI, .•• ,Nn ) is not inf div for n > 2. We shall from now on restrict attention to positive Pk
>
in (1.1). This means that we have to allow for
ak
>
I. In the next section it will become clear that we
cannot hope for more than the inf div of (1.1) for all ak
~
2.
3. THE CASE
a=2.
From (1.3) it follows that to prove the inf div of (1.1) for
ak
~
2
course, taking all
it suffices to do so for ak
equal to
2
a k =2. Of
also makes (1.1)
much more accessible for analysis. For
ak
>
2
(1.1)
cannot generally be inf div because of Lemma I.S. To see this we sketch the graph in the complex plane of the function
-
3S1 -
~
for
t
a=l
we get a half-circle).
0
in two cases: a=2
and
a
a
>
(clearly for
there exist positive
2
t l ,t 2 ,P and q=l-p i.e. there exist positive
P~a(tl )+q~a(t2)=0'
such that
2
a > 2
a "" 2
For
>
c l ,c 2 ,t o 'P and q=l-p such that P~a(cltO)+q~a(c2tO)= =0. For a=2 (or a < 2) this is not possible. We state
our findings in n
THEOREM 3 • I •
~ "-
k=1
ak
A
P k ( __ A k__ +1: )
is in general not inf
k
divif
a k >2. We now concentrate on the case
a=2,
i.e. we consid-
er (3. I)
with
Pk
>
0
(k=I,2, ... ,n)
and
0
<
AI
< ... < An.
As
our results are incomplete, we state our results with only an indication of proof. For full proofs we refer to [6J. From (3. I) it follows that n IT k=1
-
352 -
where
and, without restriction, ill S il2 S
T .=il .+iv. J J J
clear from Lemma 1.3 and Corollary 1.2 n- I' It is that in this case
S ••• S il
n
=
K' (x) F
e
~
2
-" x k
k=I
F
To prove the inf div of K'
F
~
O.
-2
-il .x n-I ~ e J cos v.x. J j=I
in (1.3) we have to show that
Sufficient from this would be that n
n-I
~
~
k=I
j=I
(3.2)
and sufficient for
-il .x J
e
(3.2) again that
S ilk
"k
(k= I , ...
... ,n-I). These inequalities however do not generally hold. By Karamata's inequality for convex e.g.
[IJ,
p.
functions
30) a weaker sufficient condition for
(see (3.2)
would be (m= I ,2 , ••• ,n - I ) •
(3.3)
It is not hard to prove, that for all and
rather than
and
m ~
n, writing
"k(n)
one has
(m=I, m=n-I),
].Jk(n)
k=I which takes care of the cases
n=2
and
For
n=3.
n=4,
(and, in fact,
a slightly stronger one) can be obtained with
some difficulty. For
n
~
5
the method seems to fail.
Using Theorem 2.6 and the identity I
2
J
J (I+Tx) g(x)dx
o
for a p.d.L
g
o
such that -
l+TX (-xg'(x))dx,
xg(x) 353 -
0
as
x
~
0
and
g(x)
0
-
as
=, we prove the inf div of (3.1) for
x -
G's having a unimodal p.d.f. Generalizing this slightly, and using the results above we have the following THEOREM 3.2. The L.T.
= J
(3.4)
o
I 2 (I+'rx) dG(x)
is inf di v i f (a) the d. f. if (b) G
has
G
is unimodal, or
or fewer points of increase.
4
The above results, supplemented by numerical evidence supporting the truth of (3.3), lead us to the following two conjectures. CONJECTURE I. If
G
is an arbitrary d.f. on
then the L.T. in (3.4) is CONJECTURE II. For
[0,=),
inf div.
> 0
nEN, A. ]
(j=I,2, ••. ,n) and
define n ~
A(z)
k=1 Let the A(z)
2n-2
zeros
z.
and
]
z. ]
(j=1,2, ••• ,n-l)
of
be ordered such that
Then m
m
~
(3.5)
ak S
k=1 for
~
Re zk
k=1
m-I,2, ••• ,n-l. REMARK. My results so far show that
arbitrary
n
and arbitrary
if m
m=1
<
m=n-I, for
or
n, and for -
n=5
354 -
if
(3.5) holds for
n-2,3 m=I,3
and or
4 4.
Of course, a counterexample to Conjecture II would not necessarily provide a counterexample to Conjecture I.
4. RELATED RESULTS Recently, interesting results have been obtained that are related to the results discussed above, which also have the following interpretation. Theorem 2. I is equivalent to the statement, first proved by GOLDIE [3J, is inf div if
that with
Y ;::: 0
ilarly,
and
Yare independent,
exponentially distributed.
and
similar circumstances
would be inf div, where
X2Y
has a Gamma distribution of order 3.2 can be used to prove that independent orders
Sim-
from Conjecture I it would follow that under
a
X and
and
a
S
X
X2
2. Finally, Theorem
XaXS
is inf div for
with Gamma distributions of
S
with
min(a,S)
~
2.
Solving three
problems posed in [7J, GOVAERTS, D'HOOGE and DE PRIL proved the infinite divisibility of for general
a
and
S
(cf.
XaXS
and
Xa/XS
[4J, also for further ref-
erences), and THORIN [9J proved the infinite divisibility of the lognormal distribution l ).
In both cases
the authors prove more than just infinite divisibility; they prove that all these distributions are "generalized Gamma convolutions". A d.f.
is called a generalized
Gamma convolution if its L.T.
is the limit of L.T. 's of
the form (a curious analogue to (I. I)) (4. I)
with canonical measure
(cf. Lemma 1.1, Corollary
1) [9J is probably the source of the same statement made by V.M. ZOLOTAREV in his lecture at this colloquium. -
355 -
1.2 and Lemma 1.3) of the form
So, generally
K'
F
is a generalized Gamma convolution if
F
F
is completely monotone, i.e. if co
J
o
satisfies
A A+T dU(A),
which is of the same form as
(2.6)
(see also Corollary
2.5) • That (4,1) is inf div is, of course, trivial.
It is
quite hard, however, to show that the distributions mentioned above have L.T. 's of this type. The authors make a heavy use of complex analysis and properties of special functions. One of the results mentioned in [4J is that p.d.L's on x
are inf dive
(0, co)
that are proportional to
a.-I (a.>0,8>0),
It would be desirable to have simpler proofs
of these very interesting results, preferably using criteria for inf div on the p.d.f. 's, which are still rather scarce. The inf div of iterated products of Gamma random variables, the distribution of which are discussed in [5J, was recently proved by L.
BONDESSON (stil
un-
published) . REFERENCES [IJ
E.F. Beckenbach - R. Bellman, Inequalities, ger, Berlin,
[2J
Sprin-
1961.
W. Feller, An introduction to probability theory and its applications, Vol.2(2-nd ed.), Wiley, New
- 356 -
York, [3J
C.M.
1971. Goldie, A class of infinitely divisible distrib-
utions,
Proc.
Soc., 63(1967),
Cambridge Phil.
1141-
I 143. [4J
M.J.
Govaerts - L.D'Hooge -
N.
de Pril, On the in-
finite divisibility of the product of two
f-distrib-
uted random variables, Applied Mathematics and Computation,
[5J
M.D.
3(1977),
127-135.
Springer - W.E. Thompson, The distribution of
products of Beta, Gamma and Gaussian random variables, SIAM J. Appl. Math., [6J
F.W.
18(1970), 721-737.
Steutel, Preservation of infinite divisibility
under mixing, and related topics, Math.
Tracts 33, Amsterdam, [7J
F.W.
Steutel,
Centre
1970.
Some recent results in infinite divis-
ibility, Stochastic Processes Appl., 1(1973),
125-
143. [8J
D.
Szasz,
Some results and problems in the limit
theory of random sums
(independent case), Limit
Theorems of Probability Theory,
Janos Bolyai, No.
ColI. Math.
II, North Holland,
Soc.
Amsterdam-
London 1975, 351-363. [9J
O.
Thorin, On the infinite divisibility of the log-
llormal distributions, Scand.
Achiavial
J.,
(1977),
121-148. [IOJ E.T. Whittaker Cambridge Univ. F.W.
G.N.
Watson, Modern analysis,
Press, London,
1958.
Steutel
Afdeling Wiskunde, Technsiche Hogeschoool Eindhoven, The Netherlands. -
357 -
COLLOQIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION METHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
MIXTURES OF INFINITELY DIVISIBLE DISTRIBUTION FUNCTIONS* S.J. WOLFE
1.
INTRODUCTION
This paper will be concerned with the following question: When are power mixtures of infinitely divisible distribution functions also infinitely divisible? Some previous work will be discussed in Section 2 and some unsolved problems will be discussed in Section 3. 2. MIXTURES OF INFINITELY DIVISIBLE DISTRIBUTION FUNCTIONS We first introduce some notation. Let
F(x)
be an
infinitely divisible distribution function with characteristic function
f(u). Let
Ft(X)
denote the
* This research was supported by Grant MCS-76-04964 from the United States National Science Foundation and by a research fellowship from the Technological University at Eindhoven,
The Netherlands. -
359 -
distribution function with characteristic function
G(x)
Let
f
t
(u).
be a distribution function with support on the
positive axis and let
G(x). Let
function of
denote the characteristic
g(u)
H(x)
be the distribution func-
tion with characteristic function h(u)
(1)
I
Let
denote the class of infinitely divisible distribu-
tion functions with support on the positive axis anrl let
V
denote the class of all distribution functions with
IF
support on the positive axis. Let
that yield an infinitely
G(x)
of distribution functions
H(x).
divisible distribution function sets, we will use the notation is properly contained in
H(x)
G
n
(x)
times is
n
A
G(x)
that if
is infinitely
is the distribution function that
has the property that the convolution of itself
Bare
is infinitely divis-
G(x)
It is easy to see that if if
and
to denote that
ACB
[I], page 427)
is infinitely divisible then divisible,
A
If
B.
FELLER has shown (see ible.
denote the class
G(x), and if
H (x) n
G
n
(x)
with
is the
distribution function with characteristic function 00
h
(u) = f f t (u) dG (t), nOn
then the convolution of
H
H(x). Thus it follows that
n
with itself
(x)
IeI -
F
divisible distribution functions Feller's theorem is not true. that if
F(x)
n
times is
for all infinitely
F(x). The converse of
In fact, HORN [2]
has shown
is a symmetric infinitely divisible
distribution function such that its characteristic function
feu)
is
log-con~ex
-
on the positive axis then
360 -
In 1971, KELKER [4]
showed that if
lCIF
distribution function then
•
F
In 1976, HUFF
a method of proof developed by ZOLOTAREV that if
F
is a normal
[13]
[3]
used
to show
is a discrete infinitely divisible distribu-
tion function with a discontinuity at the orisin, then ICIF
•
Implicit in Huff's work is the following theorem: THEOREM I. Let
F(X)
be an infinitely divisible
distribution function and assume that
N(u) (O,~)
tion of bounded variation defined on
is a functhat has the
following properties: If
1.
NI (u)
negative parts of
<
00
N2 (U)
and N(u)
denote the positive and
respectively,
then
1
f
o
2
u dN i (u)
<
for i = 1 and 2. Neither lim N.(u) = 0 ~ u- oo nor N 2 (U) are identically equal to zero.
and
NI (u) 2.
function of 3.
J Ft(X) dN(t) o
The integral
is a nondecreasing
x.
The functi on 00
(2)
g(t)
exp{J
(eiuX_I)dN(U)}
o is the characteristic function of a distribution functi on
G (x) •
Then it follows that the distribution function H(x) L~vy
defined by (I) is infinitely divisible and has a spectral function
M(u)
defined by the relation-
ship M(u) where
J F~(u)dN(t) o for
u
<
0
- 361 -
and
for
u
>
O. COROLLARY 1. If the distribution function
satisfies the hypothesis of Theorem
then
F(x)
ICI F.
At one time it was conjectured that if
is an
F(x)
infinitely divisible distribution function then
I.
IF
It now seems reasonable to conjecture that the class of infinitely divisible distribution functions
IF
have the property that
KELKER [4]
=
I
F(x)
that
is quite small.
has shown that if
is a normal
F(x)
distribution function and if
G(x)
has support on a
then
H(x)
is not infinitely
closed interval
[O,b]
divisible. This theorem follows from the fact that in this
case
has an entire characteristic function
H(x)
of the form 2
b
J o
h (z)
-
~ 2
e
dG(x)
that has zeros in the complex plane and thus cannot be a characteristic function of an infinitely divisible distribution function. This theorem can be generalized. Assume that and
G(x)
F(x)
has an entire characteristic function
has support on an interval
[O,b].
Then
H(x)
has an entire characteristic function and h (z)
Since
log f(z)
g(-i logf(z». is an entire function and
g(z)
has
infinitely many zeros in the complex plane (see [5]), it follows that
h(z)
has zeros and thus cannot be the
characteristic function of an infinitely divisible distribution function. Thus we have obtained the following
-
362 -
THEOREM 2. function
GEl
and i f
If the infinitely divisible distribution
F(x)
has an entire characteristic function then
F
cannot have support on a bounded
G
interval.
A more precise result was obtained by WOLFE for the case that [8]
is a normal distribution function.
F(x)
has shown that if
is an infinitely divisible
G(x)
distribution function then there exists a constant such that
~
I-G(x)
e
-Axlogx
for large
G(x)
has a
L~vy
A
x. This result
was also obtained independently by STEUTEL has also shown that if
SATO
[10].
SATO
spectral func-
tion with support on a bounded interval then there exists -nxlogx a constant n such that I-G(x) S e for large x. Thus his theorem gives a best lower bound for the tails of infinitely divisible distribution functions. (Both of Sato's theorems are valid for infinitely divisible distribution functions with support on the entire real line.) WOLFE
has recently obtained the fol-
[12]
lowing THEOREM 3. Let tion. that
If
GEl
I-G(x)
Thus every
be a normal distribution func-
then there exists a constant
F ~
F(x)
e
-AX (logx)
such
2
for large
GEI F , where
x.
is normal must have a
F
tail that is almost as large as the tail of an infinitely divisible distribution function. STEUTEL and KEILSON
[II]
have studied the class of
infinitely divisible distribution functions the property that if
FEF
IF = V.
then
F
that have
They have
obtained two results: THEOREM 4.
If
FEF
then
F
tion function.
-
363 -
is a symmetric distribu-
THEOREM 5. If
FEF
then
F
does not have a second
moment.
The first result follows from the fact that if
F
is not symmetric then it is possible to construct some power mixture of
that has a characteristic function
F
with zeros. The second result follows from the fact that if
F(x)
has a second moment then it is contained in
the domain of attraction of a normal distribution function. Thus if some
had a second moment it would
FEF
follow that power mixtures of normal distribution functions were infinitely divisible, a contradiction.
F.
3.
SOME UNSOLVED PROBLEMS
a)
Characterize the class of distribution functions
The class of symmetric stable distribution functions
with exponent
a
where
0
~
a
~
is contained in
F
since all of these distribution functions have characteristic functions that are log-convex on the positive axis. The set of normal distribution functions is not contained in
F.
Is any symmetric stable distribution
function with exponent in
F?
F
Can any member of
b)
<
where
a
[ 0, )]
shown that if
is contained in
x
and
XY
then any
IF
STEUTEL [9]
has
Yare positive independent
x
function with parameter random variable
FEF)
that has support on the in-
G
random variables and if
has a gamma distribution where
0
~
A
~
)
then the
has an infinitely divisible
distribution function.
G(x)
contained
Characterize the class of distribution functions
distribution function
if
2
have a finite first moment?
that have the property that i f terval
<
a
It follows from this theorem that
is a distribution function with support on
[0,11 then the function -
364 -
I
J
h (u)
(l-iu)-xdG(x)
o is the characteristic function of an infinitely divisible distribution function.
Thus the class
contains
distribution functions that have analytic characteristic functions and are not symmetric. 2 that
It follows from Theorem
cannot contain distribution functions with
entire characteristic functions. c)
Characterize the class of distribution functions
FEFO
IF = I • is degenerate at a 0
that have the property that i f
It is easy to see that if
FEFO. Does
then
FO
tion functions? Does
F(x)
then
*
contain any non-degenerate distribu-
FO
contain any symmetric distribu-
tion functions? d)
IF
Characterize the set of distribution
where
F
is a normal distribution
functions
function.
It is
interesting to compare the theorem of Wolfe with some previous theorems concerning entire characteristic functions. G
Suppose that
GEl
where
F
is normal and that
F
h (z) = has an entire characteristic function. Since . 2 ~z h(z) is entire and it follows g(-2-) . The function
from a theorem of LUKACS
[ 5]
that
have no zeros in the complex plane. a theorem of OSTROWSKI I
[6]
h (z),
and thus
g (z) ,
It then follows
that the logarithm of
must have at least order 1 and intermediate type.
from g(z)
Finally,
if -I
(3)
(x)
it follows
log log[ I-G(x)] -logx log logx
from a theorem of RAMACHANDRAN
[7]
that
lim inf (x) S I. This relation holds trivially if
-
365 -
G
does not have an entire characteristic function. follows from the theorem of Wolfe that if GEl F
lim sup x-"" I f GEl that
It then
GEl ? lim sup ~(x) ~ 1 for all F' x-co A such then is it possible to find a constant
~(x)
F
~
2.
Is
I-G(x) ~ e-Axlogx
for large
x? Can every
be expressed in the form (2) where
N(u)
GEl
F
is a function
that satisfies the hypothesis of Theorem I? REFERENCES [IJ
W. Feller, An introduction to probability theory and its applications, Vol.
[2J
London Math.
Soc.,
2(1969),
1976.
D. Kelker, Variance mixtures of normal distributions, Ann. Math.
[5J
160-162.
B. Huff, On the infinite divisibility of certain discrete mixtures, Preprint,
[4J
1971.
R.A. Horn, On certain power series and sequences, J.
[3J
2, Wiley, New York,
Statist.,
42(1971), 802-808.
E. Lukacs, Characteristic functions, Griffin, London, 1976.
[6J
I.V. Ostrowskii, On entire functions satisfying special inequalities connected with the theory of characteristic functions of probability laws, Selected Translations in Mathematical Statistics,
7(1968), 203-234. [7J
B. Ramachandran, On the order and type of entire characteristic functions, Ann. Math. 33(1962),
[8J
K.
Statist.,
1238-1255.
Sato, A note on infinitely divisible distribu-
tion functions and their Tokio Kyoiku Daigaku,
-
L~vy
measures, Sci. Rep.
12(1973), 366 -
101-109.
[9J
F.W. Steutel, Preservation of infinite divisibility under mixing and related topics, Mathematical
Centre Tracts, 33, Amsterdam,
1970.
[10J F.W. Steutel, On the tails of infinitely divisible distributions, Z. Wahrscheinlichkeitstheorie Verw. Geb., 28(1974), 273-276. [IIJ F.W.
Steutel - J. Keilson, Families of infinitely
divisible distributions closed under mixing and convolution, Ann. Math. Statist., 43(1972), 242-250. [12] S.J. Wolfe, On the infinite divisibility of variance mixtures of normal distribution functions, Nederl. Akad. Wetensch.
Proc., Ser. A,
(to appear).
[13J V. Zolotarev, Distribution of the superposition of infinitely divisible processes, Theor.
Probabil-
ity l".ppl., 3(1958),185-188.
Stephen J. Wolfe Dept. of Mathematics, University of Delaware 501 Kirkbride Office Building, Newark, Delaware, USA
-
367 -
19711
COLLOQUIA MATHEMATICA SOCIETATIS JANOS BOLYAI 21. ANALYTIC FUNCTION UETHODS IN PROBABILITY THEORY DEBRECEN (HUNGARY),
1977.
ON CHARACTERIZING THE BIVARIATE POISSON, BINOMIAL AND NEGATIVE BINOMIAL DISTRIBUTIONS E. XEKALAKI
I.
INTRODUCTION
KORWAR [IJ characterized the distribution of a nonnegative r.v.
X
as Poisson, binomial and negative
binomial when for another non-negative r.v. tional distribution of regression of
X
on
given
Y Y
the condi-
Y,
is binomial and the
X
is linear. Here we state his
result. THEOREM I. I (KORWAR [IJ). on
{O, I, ••. ,m}
Let
Y
Let
X
be a discrete r.v.
(mEI+U{+oo}). Assume that
E(x)
be another non-negative discrete r.v.
(1. I)
(0 < p < I, q=l-p, y=0,1,2, .. . ,x). Then
-
369 -
<
+00.
such that
E(xly
(1. 2)
y)
(a,b
ay + b,
-
constants)
i f and only i f (iff) Poisson (b/q) X
~
(a
{ binomial (b/I-a,
(I-a) / (I-ap»
negative binomial
(b/a-I,
(0
..
I) ,
<
a
<
I),
(I-ap)/aq) (a
>
I) •
The proof is based on the following theorem. THEOREM 1.2
(KORWAR [IJ).
Let
X,Y
Theorem 1.1. Assume that (1.2) holds. (ii)
X
ed then
is bounded i f f
0
b = m ( I -a ) , (i i i)
< 0
a
<
I.
a
<
<
be as in
Then
(i)
Also i f
X
b
>
0,
is bound-
P -I
Obviously, Theorem 1.2 ensures the positivity of the Poisson, binomial and negative binomial parameters. In Section 2 we consider an extension of Theorem
1.1 to the bivariate case which provides a characterization for the double Poisson, binomial and negative binomial distributions. A characterization of the double Poisson using RAO and RUBIN's [3J condition has been given by TALWALKER [4J. We go on in Section 3 to provide characterizations for the bivariate binomial and negative binomial distributions with p.g.£' 's (PI I + PIOs + POlt)n k
and
-k.
PI I (I - PIOs - POlt)
, respect1vely. The case of the
bivariate Poisson with p.g.f. of the form
exp[ AI (s -
is also discussed.
-
370 -
1)+
2. CHARACTERIZATION OF THE DOUBLE POISSON. BINOMIAL AND NEGATIVE BINOMIAL DISTRIBUTIONS THEOREM 2.1. Let
~
(X I .x 2 )
be a discrete random + vector on {O.I . . . . . m l } X {0.1 . . . . . m2 } (miEI U{+oo} , i=I.2). Assume that E(x i ) < +00 (i=I.2). Let ~ = (Y I • Y2)
=
be another non-negative random vector such that 2
~)
(2. I)
n
(
x.
~)
Y. x.-y. ~ p. ~ q. ~ ~
i=1 Yi
~
Y • =0. I ••••• x ., ~
~
i = I , 2) •
Then
(2.2)
E (x.
~
1Y
-
(a .• b.
}L)
~
constants,
~
i=I.2). iff
double Poisson double binomial (0
<
a.
~
double negative binomial (a.
~
>
I.
<
I,
i=I,2).
(~/(~-l);
i=I.2)
PROOF. Necessity follows immediately. For sufficiency we observe that using (2.1) and the identity
(2.3) we obtain -
371
-
-=- = -y) =
E(x.\Y
(2.4)
~
y.+(y.+I)q.x ~
~
~
y.) /(p. P(Y = y» J ~ -
(i,j=I,2, Hj).
Hence from (2.2), qi
(2.4) we have
(y.+I)P(y.
Pi
~
=
~
(a.-I)y.P(y ~
~-
y.+I, ~
=
y.) J
Y. J
-y)+b.P(y -
y)
~
(i,j=I,2, Hj).
Taking p.g.f. 's we obtain (2.5) (i=I,2)
where
Gz(~)
denotes the p.g.f. of
Z.
But-it is known (RAO [2J) that
(2.6) Then equation (2.5) can be written in terms of
Gx(~)
b.
~
(2.7)
(i)
For
a
a l " a2
at:'" log
=
I,
(i=I,2).
(2.7)
reduces to
(i=I,2)
G x(!.)
~
therefore
-
372 -
as
(2.8) From Theorem 1.2 we have
>
b.
~
0
(i=I,2).
Hence (~/~).
(2.8) represents the p.g.f. of the double Poisson For
(i i)
(i=1 ,2) we obtain from
a.*1 ~
integration
by
(2.7)
b.
b.
a .-1
a .-1
~
2
(2.9)
G
= n
(t)
X -
~
(I-a.p.) ~
i=1
~
From Theorem 1.2 if
~
~
<
a.
~
~
(2.9) represents
~
the p.g.f.
it follows
(i=1,2)
I
~
L
that b./(I-a.) is an integer and hence ~
~
{q.-(a.-I)(t.-q.)}
(~/(~-~);
of the double binomial
(~-~) / (~-ap) ) •
If, on the other hand p.g.f.
a.
~
>
I
(2.9)
(i=1,2),
is the
(~/~-l;
of the double negative binomial
(~-ap) /aq).
Hence the theorem is established. ~ =
COROLLARY I. Let in
Theorem 2.1.
Then
(2.2)
(XI ,X 2 ), !
=
holds i f f
P (x -
(y l ,Y 2 )
be as
x) -
where (i)
x.
Poisson
'U
~
(I-a .)/(I-a.p» ]
]
(i i)
(b./a.-I; ] ]
for
]
X.
a
~
=I
<
aj
,
~
(I-a.p.)/a.q.) ]
]
]
'U
binomial
a.;=I,
for
]
binomial (i*j;
'U
]
~
'U
I
Poisson (b./q.), X.
'U
~
(bi/qi)"X j
i,j=I,2),
negative binomial a.
~]
>
(i*j; ~,j
I
=
I ,2) , (iii)
X. ]
'U
X.
~
negative binomial
>
I
(i*j;
(b. / (I-a. ); ~
~
(b./a.-I; ]
]
i,]=1,2).
-
373 -
(I-a
~
) / (I-a .p . » ~
(I-a.p.)/a.q.) ]]]
]
~
for
,
3. CHARACTERIZATION OF THE BIVARIATE (DEPENDENT) BINOMIAL, NEGATIVE BINOMIAL AND POISSON DISTRIBUTIONS Before proving the main result, we need to show the following
in Theorem 2.1. Assume that for some constants b./a.-I = b./a.-I = h
a.~I,
such that
~
~
E (x. 1 y
(3. I)
(Hj)
J
a.y.+(a.-I)y.+b.,
Jl)
-
~
J
~
~
~
~
J
0
< a. <
~
(i~j,
i,j-I,2).
Then (i)
b.
>
(ii)
x
is bounded i f f
~
Moreover i f
(i=1,2),
0
is bounded then
X
(i-I ,2).
~
b.
~
(m l +m 2 )(I-a i )
(i=I,2), (iii)
0
< a. < ~
-I
Pi
(i=1 ,2).
PROOF. (i)
Letting
YI-Y2=0 equation
(3.1) becomes
Y S ~)
(since
o S E (x. 1 Y
-
~
(i=1,2).
b.
~
But equality cannot hold since it would imply that xI x 2
xiql q2
for all
P(~
x
x. But
= ~)/P(~
,Q)
= 0
i. e.
P (~
is non-degenerate. Hence
o
~)
b.
~
>
0
(i=1,2).
(ii)
Let
X
be bounded. Then from (3.1) since - 374 -
x
~
Y
we have m.
a.m.+m.(a.-I)+b.
~
~
]
~
~
(i:#j,
~
i,j=I,2),
i .e.
(3.2)
(i""1,2). From the positivity of
bi
it follows that
ai
<
1
(i=1,2). Also from (3.1) we have (3.3)
m. ~ E(x.ly = 0) ~
-
~
Hence from (3.2), So, i f
(i-1,2).
b.
-
~
(3.3) it follows that
is bounded then
X
<
0
a.
~
a.
~
The converse is also true since if for
<
1
0
<
>
(i-1,2).
0
(i-1,2). a.
~
<
1
X
were unbounded we would have from (3. 1 ) that '-
Y; $ ~
a.y.+(a.-I)y.+b. ~
~
]
~
(i:#j,
~
i,j""I,:!),
i. e. (i=1 ,2). ~
But it holds for all
only if
a.
~
~
which is
a contradiction. (iii) It has been proved that either o < a. < 1 or > 1 . In the latter case, from (3.1), (2.1) we have ~
d
i
E (x . ) ~
( I-a .p . ) ~
~
b.+(a.-I)p.E(x.) ~
~
- 375 -
]
]
(if-j,
i,j=I,2).
From the finiteness of that
<
ai
-I Pi
Also
ai
E(x.) ~
it follows
(i"'1,2)
(i=I,2).
<
I
implies
<
ai
-I Pi • Hence
0
<
ai
<
-I Pi
This completes the proof of the theorem.
(i-I,2).
THEOREM 3.2. Let in Theorem 2.1.
Then
=
(X I ,X 2 ), ~ (3.1) holds i f f X
bivariate binomial (-h; (l-a 2 )ql/c)
X'"
(l-al)q2/c,
(a i
>
I,
i=I,2)
I,
i=I,2)
bivariate negative binomial (a l -I)/QI(a l +a 2 -1), (a 2 -1)/Q2(a l +a 2 -1» (a i
(h;
<
PROOF. Necessity follows immediately. Sufficiency.
(3.4)
From (2.4) and (3. I) we obtain
P(Y.
Y.=y.)q. J
~
J
= p.[ (a.-I)(YI+Y2)
~
~
/(y.+I)P(y
-
~
~
JL)
(i ¢ j ,
+ b.J / ~
i , j = I , 2) ,
i. e. Y2- 1
P (~
(3.5)
=
Q)
hI (i,O)
n j=O
where
h i (YI'Y2)
(i,j-I,2,
i~j).
P (~ where
c
1
=
=
Yj
yi+l,
y.)/P(y J
-
Hence
= }L)
(I
P(Y i
-
r(h+Y I +Y2) YI Y2 c l r(h)Y I !y 2 ![P I (al-I)/q J [P2(a2-1)/q2J PI(al-I)/ql
-
P 2 (a 2 -1)/QZ)
(2.6) we have
- 376 -
h
.
Then from
i. e.
(3.6)
where Using Theorem 3. I it follows that if (i=I.2).
h
ai
<
is a negative integer and hence (3.6)
represents a bivariate binomial (l-a 2 )QI Ic). Also if
ai
>
I
(-h;
(l-al)Q2/c.
(i=1.2).
the bivariate negative binomial
(h;
(3.6) represents
(al-I)IQI(al+a2-1).
(a2-1)IQ2(al+a2-1». NOTE. If we allow
a.=1 ~
(i=I.2) then (3.1) reduces
to the necessary and sufficient condition for
X
to be
double Poisson (Theorem 2.1). The case of the bivariate Poisson is more complicated as the regression of
Xi
on
y
is not linear.
However. if we observe that its p.d.f. has the form
where 2FO (a.b; ;z)
=
2: a(r)b(r)zr/r! r
a(r)
=
a(a+I) •.. (a+r-l)
(r=O.I •.•• ; a(O)=I)
a characterization can be obtained as follows. THEOREM 3.3. Let
X
c
(X I .X2 ). Y = (Y I .Y 2 )
in Theorem 2.1. Then
- 377 -
be as
E(x.1 Y
(3.7)
~
- = -y)
yi+a i ZFO(-yi-I, -Y j ;; C)/ZFO(-Y I ' -Y Z;; c) (Hj,
where
<
I
c
a., ~
(i=I,Z)
x
~
i,j=I,Z)
are constants such that iff
bivariate Poisson
PROOF. The "necessary" part is straightforward. Sufficiency.
P(Y.
~
= y.+I, ~
From (Z.4) and (3.7) we obtain Y.=y.)/P(y ]
]
-
= -Y)
=
(Uj, i,j=1 ,Z).
Applying formula (3.5) we have
P <.!: where
c*
=
=
lL)
exp(-al-aZ-a1aZc).
Hence from (Z.6)
(3.8)
- 378 -
It can be easily proved using a method similar to that employed to prove part (i) of Theorem 3.1 that
>
(i=I,2). Therefore the parameters of (3.8) are positive. Hence the result. ai
0
REFERENCES [I]
R.M. Korwar, On characterizing some discrete distributions by linear regression, Comm. Stats., 4(1975),
[2]
1133-1147.
C.R. Rao, On discrete distributions arising out of methods of ascertainment, International Symposium on Classical and Contagious Discrete Distributions,
Statistical Publishing Society, Calcutta, 1963. (Also reprinted in Sankhga A, 25(1964), 311-324.) [3]
C.R. Rao, - H. Rubin, On a characterization of the Poisson distribution, Sankhya A,26(1964), 295-298.
[4]
S. Talwalker, A characterization of the double Poisson distribution, Sankhya A,32(1970), 265-270.
Miss Evdokia Xekalaki 18 Paxon St. Athens 812, Greece
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