UNBIASSED MINIMUM VARIANCE ESTIMATION CLASS OF DISCRETE DISTRIBUTIONS JOGABRATA ROY
By
Indian
IN A
and SUJIT KUMAR M...
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UNBIASSED MINIMUM VARIANCE ESTIMATION CLASS OF DISCRETE DISTRIBUTIONS JOGABRATA ROY
By
Indian
IN A
and SUJIT KUMAR MITRA
Statistical
Calcutta
Institute,
The problem of estimation of parameters involved in discrete distributions like the and Poisson Binomial and their truncated forms have been considered Binomial, Negative Most of them, e.g., Fisher (1936, 1941), Haldane (1941), Finney (1949, by various authors. 1955) David and Johnson (1952) discuss computational aspects of estimation by maximum likelihood while a few others e.g., Plackett Moore Rider (1953), (1952, 1954), (1953, 1955) of estimation which at times lead to unbiassed give other simpler but inefficient methods estimates.
In this paper, for a wide class of discrete distributions involving one unknown para minimum variance of the parameter the uniformly unbiassed estimate is (UMVU) It is shown that results for derived and the UMVU estimate of its variance is also obtained.
meter
from those for the complete disbribution by are treated as the Poisson distributions and The binomial operation. negative cases. Tables of the UMVU estimate are given for the Poisson distribution truncated
special at
at zero can be obtained
truncated
distributions a difference
zero,
for
up to
size
sample
ESTIMATES IN A CLASS OF DISCRETE DISTRIBUTIONS
1. UMVU Consider
the following prob
where
d>
ten.
0 is an unknown
discrete {X
==
probability x}
=
f(0)
=
x=
a(x)d*lf(d),
0 does not
a(x) >
parameter,
distribution
... ? a(x)0*.
defined
0, 1, 2, ..., involve
by Noak
(1950).
...
(1.1)
0 and
(1.2)
loss of generality, we shall assume a(0) ? 1. The Poisson, occur as special cases of the above. series distributions Logarithmic
Without
Negative
Binomial
and
To derive uniformly minimum variance unbiassed (UMVU) estimate for 6 on the basis = 1, ... n from of size the distribution 2, n) (1.1) we require the sample Xi(i stated below without lemmas proof.
of a random
Lemma
1.1:
If
as
tr (x) is defined tr(x)
?
=
0
a(x?r)
_v?^_
for x < r
for
r
"1 J. ...
^
x >
r
(1.3)
i J
a(x) then E{tr(X)} = Lemma
6r. 1.2:
T =
n 2 X? is a complete sufficient statistic for 6 in the sense of Lehmann
and Scheff? (1950). 371
SANKHY? :THE INDIAN JOURNAL OF STATISTICS
Vol. 18 ] Lemma
The probability
1.3:
prob
=
{T
t)
=
over
summation
denoting
= xx+x2+...+xn
ate),
values
integral
any
positive
of
..., xn
xl9 x2,
to
subject
r,
integer
ur(t) =0 C(t-r9
for
.<
r
for
. >
r
n)
C(t9 n) and noting that the probability 1.1 that Lemma
distribution
of T is of the
we may
note
that
ur(t)
...
(1.5)
same form as (1.1) it follows
from
=
E{ur(T)}
Incidentally
(1.4)
..
for
Defining
non-negative
...
0, 1, 2, ...,
of {f(6)}n and it may be expressed alternatively
C(t, w)=Sn
S
is given by
= C(t, n) &?f(6)n> t
in the expansion
where C(t, n) is the coefficient of $ as
of T
distribution
[Parts 3 & 4
.
'...
(1.6)
i-i
=
U u^t?i). ...
(1.7)
.=o
of the sufficient statistic T it follows from Rao-Blackwell (1.6) and the completeness 1947 and Rao, 1945) that ur (T) is the UMVU Theorem estimate of 6r. The (Blackwell, variance of ux(T)9 the UMVU estimate of 6 is given by From
V{ux(T)} Hence
UMVU
we
have
the
following:
Theorem
1.1:
For
estimate
because
of the variance
of ux(T)
2. Consider distribution
UMVU
are defined
prob
estimate for d is ux(T) and
the
(1.9)
...
%(%(?)-Wl(T-l)}
(1.10)
by (1.5).
ESTIMATE IN THE CASE OF TRUNCATION ON THE LEFT
the distribution
can then be
(1.1) the UMVU is
ux(T)-u2(T) ...
=
^) r =1,2,
=
(1.8)
in the form:
be put
of (1.1) may
where ur (T) for
E{ux(T)}2 -Q2....
the distribution
v(T) which
=
written {X
=
x}
on the
(1.1) truncated as =
a(x)6xlf8(6),
where f)=L(?)^,
x=
s, s+1,
... a*_=
372
left at x =
s+2,
(2.2)
...,
s?1.
The probability
...
(2.1)
UMVU ESTIMATION IN A CLASS OF DISCRETE DISTRIBUTIONS By an argument exactly = a random sample X{(i for & is ur8(T) where
to the one used
similar
1, it follows
in section
1, 2, ..., n) of size n from the distribution
for
.<
for
t^ m+r
UTi9{t) =?0 __ C8(t-r,n)
that on the basis of
(2.1) the UMVU
estimate
ns+r
(2.3)
G8(t, n)
where T = I Z? and i=l = ?.)
Q.,
S,
s (i
and xi >
over
summation
denoting
The
=
relation
following
we write For
may
CQ(t, n)
=
difference
=
operator
form
t
C8(t,n)
forr =
Complete Negative Negative
{X
=
x]
...
on G(t, x) regarded operator
(2.6)
as a function
of the integer x.
by
y)>
: ...
(2.8)
Illustrations
Binomial
Distribution. distribution:
*" (*+^1)|\K?1):
1, we have
...
Ak defined
^+i)-?WM
Binomial
=
is s ?
(2.7)
A8_1 C8^{t-n(s-l),0}. 3.
prob
(2.5)
:
A?G(t,x)\^
the bivariate
introducing
(2.5) in the alternative
complete
...
n-j),
UA C(t,n-j),
sVl)j(
A* /(?. y) =/(*+*.
form of the
=
established,
at zero, that
of truncation
in the alternative
be expressed
In general,
3.1.
...+xn
xl+x2
C(?, .i).
case of interest
the special
where A is the ordinary
can write
to
subject
I n. {a(s-l)YC8_1{t-j(s-l), )
Cx(t,n)
we
..., xn
x2,
Cs(t, n) can be easily
connecting
Cx(t,n)= which
xv
1, 2, ... n).
C,(t, n) ^(-ly where
(2.4)
i_=l
of
values
integral
II afo),...
?_
XI
0*1(1-0)-*,
373
We
x=
shall, consider
0,1,
2, ...,
the
following
(3.1)
Vol. 18 ]
SANKHY? :THE INDIAN JOURNAL OF STATISTICS
[ Parts 3 & 4
and take k as a known well-known Negative
It is positive integer as in problems of inverse binomial sampling. that the total T of a random sample of size n from this distribution has again the Binomial distribution.
prob
-
{T
.}
=
so that here C(t, n) =
Hence
for the complete Result
from
3.1:
the Negative
The UMVU Binomial
... .
(3.1)
of the variance
estimate
(3.2)
of a random
sample
of size n
is ...
T?(kn+T-1), and the UMVU
...
the following. of 6 on the basis
estimate
distribution
.
0, 1, 2, ...,
(3.3)
^-+t~W
we have
distribution
*=
t\ 0*1(1-0)'^
(^??^| (fen?1)!
(3.4)
is
of this estimate (kn-l)T
'
*
;
(kn+T-~i)2(kn+T-2)
where
T
is the
3.2.
total
Poisson
of
the
sample. truncated on the left at
distribution
zero.
For
the
complete
Poisson
distribution prob
it is well known
{X
=
x}
=
is again of the Poisson
form.
{T
=
t}
for a Poisson
=
(3.6)
of size n is given by
...
(3.7)
7^-dtlene9
=
... "?[.
(3.8)
on the left at zero, with
truncated
distribution
sample
...
Hence, C(i, ?)
Therefore
0, 1, 2, ...,
of the total T of a random
that the distribution prob
which
x=
-L xl dx?ee,
probability
law given.
by prob
we
have
which may
from
(3.3) and
be written
{X
(2.7)
=
x}^~
6xl(ee-l)9
Cx (., n)
in the usual
=
x=
~-\
a;=
1, 2, ...,
Q
form:
Ci(i.?)-^ Using
(2.3) and the results
...
of Theorem
1.1 we
thus have
374
the following:
<3-10)
(3.9)
UMVU ESTIMATION IN A CLASS OF DISCRETE DISTRIBUTIONS a Poisson distribution sample of size n from truncated on the left at zero as given by (3.9), the UMVU estimate for 6 is ux(T) and the UMVU where estimate for the variance of u?(T) is V(T) = ux(T) {u^-u^T-l)} Result
3.2:
of a random
On the basis
ux(t)
=
tA^O1'1 ?_"0r
T being
The values
for t ^
n+l
of u?(t) correct to three places of decimals for n = of zero tabulated by Fisher and Yates (1933) may to be less than 25.
2 gives values of the differences
10 if T happens
Plackett
n+l
the sample total. Table
n>
for t<
0
On the basis of a sample xv x2, ... xn from a Poisson an unbiassed estimate (1953) suggested -
2
n
iml
= yi
where
= The variance
of this estimate
0
distribution
if Xi =
x,i if xi >
for
truncated
at zero
relative
to the
1 "? 1. J
is
estimate
1. EFFICIENCY OF PLACKETT'S ESTIMATES RELATIVE ESTIMATE BASED ON A SAMPLE OF SIZE 10 variance Plackett's estimate
This
..., 10.
_/.
of Plackett's The following table (Table 1) shows the efficiency ? 10 and several values of 6. UMVU estimate for n
TABLE
2,3,
be used
of
UMVU
-efficiency
(3)/(2)
estimate
(1)
(2)
(3)
1.0
0.158198
0.151703
0.9589
2.0
0.262607
0.252182
0.9603
3.0
0.347156
0.338585
0.9753
4.0
0.439852
0.424500
0.9875
shows that Plackett's
estimate
TO UMVU
is highly
is a slight fall in efficiency.
375
(4)
efficient.
For
larger sample
sizes there
Vol.
SANKHY? : THE IND?AN JOURNAL OF STATISTICS
18 ]
[ ?arts 3 & 4
TABLE 2. UMVU ESTIMATE OF THE PARAMETER IN POISSON DISTRIBUTION TRUNCATED AT ZERO (BASED ON A SAMPLE OF SIZE n WHEN THE TOTAL OF THE OBSERVATIONS IS t) t
n234
5 ?
o--
2 3
1.000
4
1.714
?
?
?
o
1.200
0.500
0
2.903
1.667
0.923
0.400
3.444
2.093
1.300
0.750
5
2.333
6 7
?
?
?
__
__________ 0 0.667
10
6789
?-
?
?
?
?
?
?
?
?
0 0.333
? 0
r
?
? ?
8
3.969
2.493
1.646
1.067
0.632
0.286
9
4.482
2.874
1.970
1.360
0.950
0.545
0.250
0
0
10
4.990
3.242
2.278
1.635
1.159
0 786
0.480
0.222
11
5.495
3.601
2.574
1.399
1.011
0.694
12
5.997
3.953
2.860
2.146
1.627
1.224
0.9
13
6.498
4.299
3.139
2.388
1.846
1.427
14
6.999
4.642
3.412
2.623
2.057
15
7.500 *
4.983
3.680
2.852
2.262
* For
values
16
1.896
0.419 6
0.200
0.622
0.387
1.088
0.805
0.564
1.622
1.272
0.980
0.731
1.810
1.448
1.147
0.892
5.321
3.945
3.076
2.461
1.993
1.618
1.309
1.046
175.658
4.206
3.297
2.656
2.170
1.783
1.464
1.194
5.994 18
4.466
3.514
2.847
2.343
1.944
1.615
1.338
196.329
4.723
3.728
3.034
2.513
2.101
1.762
1.477
206.664
4.979
3.940
3.219
2.679
2.254
1.096
1.613
21 6.998
5.233
4.150
3.401
2.843
2.404
2.046
1.746
22
5.487
4.358
3.582
3.004
2.552
2.184
1.875 2.002
7.332
23
7.666
5.740
4.565
3.760
3.163
2.697
2.319
24
7.999
5.992
4.771
3.937
3.320
2.840
2.451
2.127
25
8.333 * 26
6.244
4.976
4.112
3.476
2.982
2.582
2.249
of t beyond
6.495
5.180
4.286
3.630
3.121
2.711
2.370
276.746
5.384
4.460
3.783
3.259
2.838
2.489
286.997
5.586
4.632
3.934
3.396
2.964
2.606
297.248
5.789
4.803
4.085
3.532
3.088
2.722
307.498
5.991
4.974
4.235
3.666
3.121
2.836
31
7.749
6.192
5.145
4.384
3.800
3.334
2.950
32
7.999
6.394
5.314
4.532
3.933
3.334
2.950
33
8.249
6.595
5.484
4.679
4.065
3.575
3.173
34
8.499
6.796
5.652
4.826
4.196
3.694
3.284
35
8.750 *
6.996
5.821
4.973
4.327
3.813
3.393
7.197
5.990
5.119
4.457
3.931
3.502
37 7.398
6.158
5.265
4.586
4.049
3.610
387.598
6.326
5.410
4.715
4.166
3.717
397.798
6.4?.4
5.555
4.844
4.282
3.824
407.999
8.661
5.700
4.972
4.398
3.930
this
use
36
t\n.
376
UMVU ESTIMATION IN A CLASS OF DISCRETE DISTRIBUTIONS 2
TABLE n
56
(Contd.)
7
8
9
10
41
8.199
6.829
5.845
5.100
4.513
4.036
42
8.399
6.996
5.989
5.228
4.628
4.141
43
8.599
7.163
6.133
5.355
4.743
4.246
44
8.799
7.331
6.277
5.482
4.957
4.351
45
9.000 *
7.498
6.421
5.609
4.971
4.455
7.665
6.565
5.736
5.085
4.557
7.832 47
6.709
5.862
5.199
4.662
7.999 48
6.852
5.989
5.312
4.765
8.165 49
6.996
6.115
5.425
4.868
8.332 50
7.139
6.241
5.538
4.971
8.499 51
7.282
6.367
5.651
5.073
8.666 52
7.426
6.493
5.763
5.175
8.833 53
7.569
6.619
5.867
5.278
46
8.999 54
7.712
6.744
5.988
5.379
9.166 55
7.855
6.870
6.101
5.481
*
56
7.998
6.996
6.213
5.583
8.141 57
7.121
6.325
5.684
8.284 58
7.246
6.437
5.786
59 8.427
7.372
6.549
5.887
8.570 60
7.497
6.660
5.988
8.713 61
7.623
6.772
6.089
8.856 62
7.748
6.884
6.190
8.999 63
7.873
6.995
6.291
64 9.142
7.998
7.107
6.392
9.285 65 *
66
377 23
8.123
7.218
6.492
8.249
7.330
6.593
678.374
7.441
6.694
8.499 68
7.553
6.794
8.624 69
7.664
6.895
8.749 70
7.776
6.995
8.784 71
7.887
7.096
8.999 72
7.998
7.196
739.124
8.109
7.296
9.249 74
8.221
7.397
9.375 75
8.332
7.497
*
8.443
7.579
778.554
7.697
788.666
7.798
798.777
7.898
808.888
7.998
76
Vol. 18 ]
SANKHY? :THE INDIAN JOURNAL OF STATISTICS . TABLE
2
n
[Parts 3 & 4
{Contd.)
9
10
8.999
8.098
t 81 82 83 84 85 86 87 88 89 90
9.110
8.198
9.222
8.299
9.333
8.399
9.444 *
8.499 8.599 8.699 8.799 8.899 8.999
91 92 93 94 95 96
9.099 9.199 9.299 9.399 9.500
Refekences D.
Blackwell, Stat., F. N.
David, Finney,
D.
Finney,
D.
and
Johnson,
A.
13-25. The
-(1941): A. and
Lehmann,
E.
S.
L.
P. G.
(1952):
effects
of method
Ann.
Biometrics, 14,
Eug.,
328.
Poisson
truncated
175-185.
8,
319
Math.
Ann.
estimation.
sequential
Poisson.
of the
example
distribution.
Biometrics,
of frequencies.
Ann.
of Binomial (1950):
Ann.
182-187.
11,
Eug.
for Biological,
Eug.
and Medical
Agricultural
Research
1953.
Edinburgh,
fitting H.
10,
Tables
estimation
upon
Distribution.
Statistical
& Boyd,
Scheff?,
of ascertainment
Binomial
(1953): The
Sankhy?, The
distribution
Ann. Similar
Completeness,
179-1S1.
11,
Eug.
and unbiassed
Regions
estimation.
305-340. of the Poisson
estimation
from
parameter
a truncated
distribution.
Biometrika,
247-251. A
-(1954): (1950): R.
Plackett, C. R. Bull.
Paper
(1955):
F.
(1941):
I.
P.
An
G. C.
Oliver
and
Part
39,
Rider, _-.?.-
The truncated (1952): Binomial distribution.
L.
Negative
Yates
Edition,
J. B.
Rao,
The
(1936):
Haldane.
Noak,
unbiassed
387?394.
6,
A.
and
expectation
truncated
(1949). J. and Varley,
Fourth
Moore,
N.
The
R.
R
Fisher
105-110.
J.
11, Fisher,
Conditional
(1947): 18,
R.
L.
A
on
note class
The
(1945):
(1953): Information
Cal.
Math.
37,
(1955):
Soc, Truncated
Truncated
received : January,
and
Poisson accuracy
distribution.
with
variables
truncated
(1953):
Poisson
truncated
of random
discrete distribution.
Ann.
the
Math.
Stat.,
21,
127-132.
9, 485-488.
Biometrics, in
attainable
402-406.
10,
Biometrics, distributions.
estimation
of
statistical
parameters.
81-91. Poisson
Binomial
distributions. and
Negative
1957.
378
J.
Amer.
Binomial
Stat.
Ass.,
distributions.
826-830.
48, J.
Amer.
Stat.
Ass.,
50,