•
BOUNDED RISK ESTIMATION OF A FINITE POPULATION MEAN:
OPTIMAL STRATEGIES
by
Nitis MUkhopadhyay
p?'anab Kumar Sen
Bikas Kumar Sinha
"
•
Institute of Statitics Mimeo Series No. 1812
December 1986
BOUNDED RISK ESTIMATION OF A FINITE POPULATION MEAN:
. OPTIMAL STRATEGIES
•
..
NITIS MUKHOPADHYAY, PRANAB KUMAR SEN AND BIKAS KUMAR SINHA*
•
•
I
I
ABSTRACT
•
For the mean of a finite population, a bounded risk estimation problem is considered for both the situations Where the
population variance mayor may not be known.
In this context,
three popular (equal probability) sampling strategies are considered.
These are the analogues of (i) simple random sampling
with replacement, mean per unit estimation, (ii) simple random
sampling with replacement, mean per distinct unit estimation, and
(iii) simple random sampling without replacement, mean per unit
estimation.
It is well known that in the conventional fixed-
sample size scheme, (iii) fares better than (ii) and (ii) better
than (i).
However, in the current context, the sample sizes are
dictated by (possibly, degenerate) stopping times, and visualizing
the cost (due to measurements/recording, etc.) as a function of
the number of distinct units in the sample (as pertinent to
schemes (i) and (ii»and identifying that in scheme (iii), the
number of distinct units is equal to the sample size itself, we
are able to show that the second strategy still fares better than
the first, although the third strategy may not perform better
than the second one.
•
Actually, in the case of known population
variance, it is shown that in the light of the number of
distinct units, the ASN (average sample number) for the second
strategy can never be greater than two plus the ASN for the third
strategy and can never be less than the latter minus one.
similar relationship also holds in the case of unknown
A
2
population variance When we define the stopping rules in a
coherent manner.
Interestingly enough, this is quite contrary
to our age-old belief that simple random sampling with replacement
can never perform better than simple random sampling without
replacement.
Our theoretical results are backed up by numerical
examples, too.
Also, dominance of Strategy (ii) over (i) in a
general sequential setup constitutes an important task of the
current study.
Finally, to reconcile Strategies (ii) and (iii)
in a general sequential setup, the coherence of the associated
stopping times has also been discussed thoroughly.
KEY WORDS:
ASN; Distinct units; Minimum risk; Sequential
estimation; Simple random sampling; With replacement; Without
replacement.
•
3
*Nitis Mukhopadhyay is Professor, Department of Statistics,
•
University of Connecticut, Storrs, CT 06268; Pranab Kumar Sen
is Cary C. Boshamer Professor of Biostatistics and Adjunct
Professor of Statistics, University of North Carolina, Chapel
Hill, NC 27514; and Bikas Kumar Sinha is Professor, Indian
Statistical Institute, Calcutta 700 035, India.
He is currently
visiting the University of Illinois at Chicago, IL 60680.
This
work was partially supported by (i) Office of Naval Research,
Contract N00014-85-K-0548, and (ii) Office of Naval Research,
Contract N00014-83-K-387 •
•
4
1.
INTRODUCTION
We consider a finite (labelled) population of N units,
serially numbered 1, .•• ,N.
Denote by Y the study-variate which
assumes values Y ' •.• 'Y on the units 1, ••• ,N, respectively.
N
1
finite population mean (y) and variance
-1 N
Y = N l:i=lY i
(0 2 )
The
are defined by
and
(1.1)
Also, for later use, we write
(= N(N-l)
-1 2
0 ).
(1.2)
We are primarily interested in the estimation of Y (i.e.,
the finite population mean) with a bounded risk.
problem,
0
2 mayor may not be known.
In this
Also, for this problem, we
may consider the following sampling strategies (as extended to
the sequential case, whenever needed):
(i)
Simple random sampling with replacement (SRSWR), mean
per unit estimation;
(ii)
(iii)
SRSWR, mean per distinct unit estimation;
Simple random sampling without replacement (SRSWOR),
mean per unit estimation.
In the conventional fixed-sample case, a relative comparison
of the above strategies is' well known [viz., Basu (1958), Raj
and Khamis (1958) and Asok (1980), among others].
are known to be progressively better.
The strategies
However, in the current
context, the results seem to indicate that while the ordering
5
between the analogues of the first and second strategies remains
the same, the ordering between the analogues of the second and
third strategies may change in some cases.
This is contrary to
the popular belief that SRSWOR always performs at least as good
as, or better than, the SRSWR.
To set our analysis in the proper perspective, in a SRSWR,
we denote the random variables and indexes associated with the
successive drawings by (Yk,r ), k
k
r
k
Yk
~
1, so that for each
takes on the values 1,.:.,N with equal probability N-
= Yr
k(~
l
1),
and
Then, the simple mean per unit estimate of Y
, k > 1.
k
(based on a sample of size n) is given by
= n -1 L.n1= 1Y1.
(1.3)
It is well known that Y is an unbiased estimator of
n
Yand
(1.4)
Let us then consider a sequence
{~;
k > 1} of indicator
variables, where
I
k
={
with 1
Then, for every n
vn =
n
rk=l~
1'
otherwise;
0,
1
~
k
>
1
(1.5)
= 1.
1,
denotes the number of distinct units
in the sample of size n.
(1.6)
6
Note that v l • 1 , vn is
E(v )
n
1 in n, and
= N{l
- (1 - l/N)n),
Y n > 1.
U.7>
The mean per distinct unit (in the sample of size n) is given by
U.8)
Note that y(v ) is also unbiased for
Yand
n
2
N
1
1
• a (-){E(-)
- -N},
N-1
v
(1. 9)
n
In SRSWOR, for n sample units, the indices are denoted by R1 ,
..• ,R, so that R
n
-n
=
(Rl, ... ,R ) takes on any (unordered) subset
n
of n out of N numbers {l, •.. ,N} with the same probability
N-[n]
= {N
.•. (N - n + l)}-l, and the sample random variables
,
= y~,
Yl'···'Yn are given by Yk
k =l, ... ,n.
The mean per unit
,.
estimator is
-'
Y
=
n
n
-1 n
'
Li=lYi ' n ~ 1.
(1.10)
Like the other two estimators, Y is also unbiased for Y and
n
-' - -y) 2
E(y
n
= S2 (-n1
1
- -).
N
0.11)
Note that in the SRSWR, whenever a unit is chosen in the
sample for the second time (or later), we do not incur any cost
•
for its measurement (or recording),
and, hence, it seems quite
plausible to have a cost function c(n)
is a constant.
= cv
, n > 1, where c(> 0)
Thus, c(n) is stochastic in nature and
n
7
(1.12)
In SRSWOR, v
n
= n,
and hence, c(n)
= cn
is non-stochastic.
This
difference in the nature of the cost function plays a basic role
in the sequential schemes to be considered here.
We may consider, for an arbitrary estimator T
n
usual squared error loss function A (T
a given positive constant.
n
of
Y,
the
- y)2, where A(> 0) is
Then corresponding to a
~iven
upper
bound We> 0) to the risk of any estimator of Y, we may define
nO
= min{n
~
1 : AE(T
- 2
n
- y)
~
(1.13)
W}.
We may then compare the Ec(n ) for different sampling strategies.
O
This is what we may call the bounded risk approach for the
comparison of the different sampling strategies.
noted that generally E(T
n
It may be
- y)2 involves the unknown 0 2 (or S2),
and, hence, we may need to consider suitably modified stopping
rules which, of course, would generally make the analysis more
complicated.
This aspect will be studied in detail in Section 3.
An alternative approach to (1.13) would be to consider the risk
function
p (T ,Y)
n
n
= AE(Tn
- -2
y) + Ec(n)
and to determine n in such a way that (1.14) is a
(1.14)
~inimum.
Then, it seems quite plausible to compare these "minimum risks"
for the different strategies.
approach is needed when
02
Here also a "stopping rule"
(or S2) is not known.
We shall
•
8
mainly confine ourselves to the first (i.e., bounded risk) approach,
and indicate how parallel results hold for the "minimum risk"
approach.
2.
BOUNDED RISK ESTIMATION OF
Y:
0
2 KNOWN
Consider the following strategies:
Strategy I:
Keep in mind (1.3) - (1.4).
sampling and use Y
n
o
as an estimator of
Y,
Adopt (SRSWR, nO)
where nO is so chosen
that
AW- l 0 2 ~ nO
fo~
Note that by virtue of (1.7),
an estimator of
AW- l 0 2.
(2.1)
this strategy, we have
-1 nO
= cN{l
Ec(n ) = cE(v )
O
nO
Strategy II:
< 1 +
- (1 - N )
}.
(2.2)
Adopt (SRSWR, n*) sampling and use y(v *) as
n
where y(v ) is defined by (1.8) and n* is so
Y,
n
chosen that
E{__l__} < ! + ~
vn* - N AS 2
<
E{
1 }
vn*-l
(2.3)
the motivation for this choice of n* is derived from (1.9).
For
this strategy, we have
Strategy III:
Keep in mind (1.10) - (1.11).
,
(SRSWOR, n**) sampling and use
•
J
Yn**
Adopt
as an estimator of Y, where
n** is so chosen that
l/n**
1
< -
- N
W
AS2
+ --- < l/(n**-l).
(2.5)
9
In this case. we have
Ec(n**) ... cn'ir*.
Clearly, for known
0
(2.6)
2
2
(or S ), each of the strategies leads
Ywith
to an unbiased estimator of
"bounded risk", and hence a
comparison of (2.2), (2.4). and (2.6) would reveal the relative
efficiencies of these strategies.
We term (2.2). (2.4). and (2.6)
as the Average Cost Funation of Strategies I, II, and Ill,
respectively, and denote them by ACF (I), ACF(II), and ACF(III)
in that order.
Note that these are all functions of A, W, 0 2 and
N.
Theorem 2.1.
Unifo1'17lZy in
A~ W~ N~
and
02 ,
ACF(II) < ACF(I).
Proof·
(2.7)
First. we may note that [c.f., Asok
E(
l.)
vn
1
N-1
N -< --Nn
- -
,
Y
n _>
(1980)]
1,
(2.8)
so that
1
E{v
Writing B
= N-1
2 -1
+ W(AS)
B
Since S2
= N(N
n*-l
<
}
N+n*-2
~ N(n*-l) •
(2.9)
,we have from (2.3) and (2.9),
(N+n*-2)/{N(n*-1)}.
(2.10)
- 1)-1 02 , (2.10) and some routine steps lead us to
n*
< 1 +
AW-1 0 2 .
(2.11 )
Thus, by (2.1) and (2.11), we have
n* < nO + 1. '
(2.12)
10
and as n* and nO are both positive integers. we have. therefore,
Consequently, by (2.2) and (2.4). we have Ev *
n
and this implies (2.7).
Theorem 2.2.
ACF(II) - ACF(III)
<
nO
Q.E.D.
Unif01'1TlZy in A~ W~ N and
- c
Ev
<
-
02•
2c.
<
(2.13)
Before we proceed to prove this theorem. we may note that
(2.13) actually relates to the inequality:
-1 < E(V *) - n**
n
<
. 2
2, uniformly in A. W. Nand 0 .
(2.14)
Or, in other words. E(V *) cannot be smaller than n** - 1
n
n** + 2.
and also it cannot exceed
We shall show by some
numerical examples that EVn* may be sometimes less than n**.
while it may also be greater than n**.
The major implication of
this theorem is that Strategy III may not always perform better
than Strategy II; they are generally very "close" in their performance characteristics.
Lemma 2.3.
For SRSWR (N.n), for every
(Ev )-1
n
Proof.
In this context. we need the following.
<
-
E(V- l )
n
<
(EV
_ )
n l
n > 2
-1 .
(2.15)
Note that [viz., Chakrabarti (1965), Korwar and
Serfling (1970), Pathak (1961)]
,n-1
('/N)n-1
E(v- l ) = N-nL~
= N-1l:~
J=1 J
n
J=1 J
1
= N- {1 + L~-1('/N)n-1}
J=l J
"
.. N-1U + L::~(l-k/N)n-l} •
(2.16)
11
On the other hand, by (1.7),
Ev
n-l
~ N{l - (1 - l/N)n-l}, n
> 2.
(2.17)
Therefore,
(2.18)
>
N-l{l + EN-l(l _ N- l )k(N-1)}
k=l
•
Again, it follows easily (by induction on k
~
(2.19)
1) that
By (2.16), (2.18), and (2.20), we immediately get that
(EV _ )
-1
n l
On
the other hand, E(v )E(v
n
-1
n
E(v
> EV
-1
n
•
(2.21)
) > 1, so that
-
-1
-1
) > (Ev) .
n
n
(2.22)
Thus, (2.15) follows from (2.21) and (2.22).
Proof of Theorem 2.2.
By (2.3) and (2.5),
(2.23 )
while by (2.15) and (2.23),
(2.24)
Note that (2.24) ensures that
EV n *_2
<
n**.
(2.25)
12
On the other hand, by (1.7)
EV n
=1
+ (1 -
~)
1 2
N
n-2'
+ (l - -) Ev
(2.26)
so that by (2.25) and (2.26), we have
Evn*
<
1 + (1 - ~) + (1 _~)2n**
+ (1 - N- 1n**)(2 - N- 1 ).
= n**
(2.27)
Further, by (2.3), (2.5), and (2.22),
(2.28)
so that
Ev
n*
>
n** - 1.
(2.29)
Combining (2.27) and (2.29), we have
1
1
< n** + (l-N- n**)(2-N- ) < n** +2
(2.30)
n*
for every N, W, A, and S2. This completes the proof of Theorem
n** - 1
2.2.
Ev
<
Q.E.D.
Remark 1:
It may be noted that in the above analysis, choice
of W is quite arbitrary and it is generally left to the experimenter.
Two particular choices based on cost considerations may
be suggested.
(a)
= risk
2
-1
-1
Choose n** beforehand and set W = AS {(n**)
- N }
l
attained by the use of {SRSWOR(N,n**), -'
Yn**} strategy.
For the competing strategy {SRSWR(N,n*), Y(v *)} with the same
n
bound W to the risk, the expected sample size E(V *) satisfies
n
l
'"
the inequality
n**
<
E(Vn*)
<
n** + (1 -
n**
~)(2
-1
- N ).
13
(b)
Choose n* beforehand and set W
2
= AS 2{E«V n*)-l)
- N- l }
~ risk attained by the use of {SRSWR(N,n*), y(v *)} strategy.
n
I
Then determine n** such that the use of {SRSWOR(N,n**),
strategy yields the same bound W to the risk.
2
Yn**}
This time we can
prove a slightly improved version of (2.30), viz.,
n** - 1
<
E(V *)
n
Next, recall that B
< 1 +
=1
+
N
then, of course, n**
On
B
-1
= ex
= B-1 ,and
1
(1 - N- )n**
<
n** + 1.
W so that if B- 1 is an integer,
AS2
) > n**.
-1.
.
d wh en B 1S not an 1nteger, let us set
the oth er h an,
o
+ S,
<
S
<
=
1, ex
hence E(v
n*
[B- 1 ], the integral part of
B
-1
•
-1
1 + ex while E(v *) > B = ex + s. Thus, at least
n
for small values of S, there is a possibility of E(V *) being
n
Then, n**
I:
smaller than n**.
This is indeed true in some cases as
evidenced by Table 1.
[Table 1 goes approximately here.]
Remark 2:
-1
If W ~ 0 so that B
~
N, ex becomes large and,
hence, we can expect that for a wider range of S-va1ueA, E(v
~)
n"
would be smaller than n**.
We conclude this section with some CODDllents on the "minimum
risk" approach.
-
2
Note that by (1.4), (1.9), and (2.8),
-
-
2
E(y(v ) - y) ~ E(Yn - y) , V n ~ 1, and, hence, noting that
n
Ec(n) is the same for both Strategies I and II, we conclude that
y
14
so that Strategy II fares better than I.
To compare Strategies
II and III, we note that if n** is the specific value of n for
which
p
n
-' , Y) is a minimum, we have
(y
n
AS2(n**(~**+1»
<
-'
c
-
~ AS2(n**(~**_1».
1
= AS 2 (n**
On the other hand, inf Pn(Yn' y)
(2.B)
1
- N) + cn**.
n
Therefore, we have
l 2
c(2n**-1) - N- AS
inf
<
Pn(Y~'Y) ~ c(2n**+1) - N- l AS 2 •
n
(2.e)
the other hand, suppose that Pn*(Y(v )'Y)
n*
Then
On
(2.D)
where Ev n* need not be a (positive) integer, while n** is so.
Thus, whenever n** - 1
<
EV *
n
<
n** + 1, but
(A/c)~S
is not an
integer, the right-hand side of (2.D) may actually be smaller than
Pn**(Y~**'Y).
smaller than
However, if n**
Pn**(Y~**'Y)'
=
(A/C)\S, then (2.D) cannot be
the true minimum of
AS2(~
-
j) + cu,
over u > O.
In other words, a lower bound for P *(Y(
),Y) may
n , v n*
not necessarily be greater than or equal to Pn**(Yn**, Y). On
-1
"
the other hand, Ev * = 1 + (l-N )Ev *_l and by Lemma 2.3,
n
n
-1
-1
Ev n* -< (EV n*_l) • Thus, we have
15
<
-' ,Y), for all m, we may even
Pm(y
\I
m
use a crude upper bound:
c{E\I +(l-N -1 E\I)}
m
m
for an arbitrary m.
We choose m such that n**
<
E\I
m
<
(2.E)
n** + 1.
Then from (2.E), we have
-
Pn* ( y(\I
-)
),Y
n*
~ AS
2
II}
{;i*
- N
+
C
{
n** + 1 + 1
_ ~}
N
(2.F)
(2.F) is comparable to (2.30).
-'
-
as c + 0, while Pn**(Yn**'Y)
Note that by (2.B), n**
= 0 e (c~).
= 0 e (c-~)
Thus, by (2.F), we
conclude that as c + 0,
(2.G)
This clearly depicts the "closeness" of the two minimum risks for
the Strategies II and III.
3.
BOUNDED RISK ESTIMATION OF Y:
0
2 UNKNOWN
For the case of infinite population, sequential procedures
for this problem were considered by Robbins (1959), Chow and
Robbins (1965), Ghosh and Mukhopadhyay (1979) among others.
Along their lines, we may consider the following (modified)
strategies.
Strategy I':
Sample units one by one at random and with
replacement, governed by the stopping rule:
16
TO • min{n
Here, s2
n
=
~ 2: n ~
(n-l)-l r~ ley. 1=
1
arbitrary positive constant.
(sequential) estimator of
Strategy II':
Yn )2
-1
2
-y
W A(sn + n )}.
0.1)
for every n > 2 and y is an
-
Then, YT
is the desired
o
Y.
Sample units one by one at random and with
replacement, governed by the stopping rule:
N-\I
T* = min{\I > 2 : \I > W- 1A( n s2(\1 ) + \In-Y)}.
n -
n -
N
0.2)
n
2
Here, s(\I ) is the sample variance based on \In distinct units
n
(with division \I -1) and y is an arbitrary positive constant.
n
Consider
Y(T*)' the mean per distinct units, as the (sequential)
estimator of
Y.
Strategy III':
Sample units one by one at random and with-
out replacement along the stopping rule:
0.3)
2
Here, sand yare, respectively, the same as they were in (3.1).
n
Consider
Y(T**) as the (sequential) estimator of
In the case of
02
Y.
unknown, whichever strategy is adopted,
the sample size is a random variable, and, consequently, the
properties of the estimatQrs of
Ymay
change.
As a matter of
fact, it is not difficult to verify that none of the above
estimators remain unbiased in a general setup.
Thus, it may be
more pertinent to compare EC(T )' CE(T*) and CE(T**) along with
O
17
the mean squared errors (MSE) of the sequential estimators Y '
T
-y(T*)
o
-'
and Y(T**)·
Regarding Strategies I' and II', the former involves the
sample variance based on all the observations, while the latter
is based on
s~v ) (i.e., on the distinct units only), although
n
both are adapted to the SRSWR.
Bahadur (1954) has pointed out
that in sequential decision problems, attention can be confined
to sequential decision (including stopping/action) rules which
depend at each stage, n, on a transitive sufficient sub-sigma
field B* whenever the latter exists.
n
We show in the Appendix
that under SRSWR (in a sequential setup) there exists a minimal
sufficient transitive sequence {B* , n > l} based on the distinct
n
units in the sample.
-
This means that in the definition of the
stopping rule, attention can be concentrated on the use of v
and
s~v
)' instead of the
n
more relevant.
s~.
n
Consequently, Strategy II' is
Thus, we would advocate the use of II' instead
of I'.
Next, we come to the comparison of Strategy II' and III'.
Let us examine the two stopping rules T* and T** in (3.2) and
(3.3).
variable
Recall that v
n
«
n) is a positive integer valued random
(in a SRSWR), while n in a SRSWOR is non-random.
However, using Hajek's (1964) rejective sampling (equal probability)
scheme, we may equivalently reduce the SRSWOR to SRSWR with distinct units
18
only if we consider a sequence {M , n > I} of integer valued
n
random variables, defined by
M
n
•
= min{k
>
1
\l
-
As such, we may write -'
yn....
D y( \I
M
k
= n},
(3.4)
n > 1.
) for all n -> 1.
Here, we
n
write U ~ V to mean that the random variables U and V have
identical distributions.
A similar distributional identity
holds for s~ (in SRSWOR) and S~\lM).
Consequently, by (3.2) and
n
(3.3), we may conclude readily that
T* =
D T** and
y (.*) =
D -'
y (.**)'
(3.5)
so that Strategy II' and III' share the common properties.
This
feature is not surprising, as in (3.2) and (3.3) we have used
•
essentially the same stopping rule.
In the case of
the situation was slightly different [as
\1
Mn
=n
was not].
\I
n
0
2
known,
was random while
Looking at (2.3) [and (2.16)], we may as well
consider a modified stopping rule
0.6)
and propose the distinct mean estimator y(.*) for
o
Y.
The stopping
rule in (3.6) may still be motivated by the transitive sufficiency
of {B * , n
n
~
I} based on the distinct units (in SRSWR) and,
apparently, this is more in line with (2.3) [than (3.2)].
With
this stopping rule in (3.6), the distributional equivalence results
in (3.5) do not hold (when
.*
*
is replaced by '0).
However, as in
19
the case of
0
2 known, here also
.**
(or -'
y.**) may not aLways
dominate '*0 (or y *). Towards this, we consider the following
'0
numerical example which shows that we can simultaneously realize
(i)
In other
E(v *) < E(.**) and (ii)
'0
words, we demonstrate that
.~
.**
may indeed be better than
in
some cases.
ExampLe:
We take N
=5
= 0,
Y2
and choose A = W, Y = 1.
= 1,
Y3
= 1.2,
Strategy III': Stopping RuLe
.*
in (3.3).
variate values be Y1
so that Y
Y4
= 1.4,
Let the
Y
5
= 2.5
= 1.22.
{(i j)!l ~ i
r
Samples:
j ~ S except (IS) and (51)}, (lS2), (lS3), (154),
(S12), (S13) and (S14) where i,j, etc., refer to the labels of
the sampled units.
-
E(y~**
-2
- y)
=
Now E(.**)
=
2.10, E(Y~**)
= 1.2183
and
.240814.
Strategy II':
Stopping RuLe
'0
in (3.6).
{(i j),(iij),(iiij), ..• for all (ij),
1
r
Samples:
j, except (S 1) and
(1 S)}; (1 S 1), (1 S 2), (1 S 3), (1 S 4), (1 5 5), (S 1 1),
(5 1 2), (S I 3), (5 1 4) and (5 1 5).
Now,
E('~)
= 2.048,
E(y *) = 1.2192 and E(y * - y)2 = .240697.
'0
'0
We hereby conclude that .~ provides smaller Bias, ASN and MSE
compared to .**.
20
APPENDIX: MINIMAL SUFFICIENCY AND TRANSITIVITY IN SRSWR
Let N
Yn
n
= n-1 1:.u:Nf n1•Y.1 ,
f
= r.1£Nf n1.,
= ~f
of times the index i appears in
the sample S = (sl,··o,sn)
n
= A( fn1 '
f nN) , n > 1 ( increas ing) .
ni
A
n
0
•
0
,
Let
gni
"n
y(" )
n
N
•
N
nO
if f
of'
0, i f f
> 1
ni
=0
ni
= Ei£JIlgni «
,
1
<
n
= " n-1 1:.1,£Ngn1.Y.1
B
= NnO
(E. N
= {i
£
N
Cardinality of Nn 1
n
Nnl
U
g
N
= B(gnl,···,gnN)'
B
n)
i
<
(.
A
n'
f.
1£ n1 n1
=
ni
= "n ,
ltd n > 1.
= n)
OJ,
Cardinality of Nn 0
* = B("n ,Nn 1,Nn 0)
n > 1 (,)
=N -
"n .
n c. An ).
B
(<:: B
n
Note that conditional on B* , the joint probability function of
n
fn
=
(fnl,o.o,fnN ) remains invariant under any permutation of the
"n indices {i £ N } among themselves and N - "n indices {i £ N }
n1
nO
among themselves.
Therefore,
21
E{y
n
IB}
* = n -1 r.1ENY.E(f
. IB* )
n
1
n1 n
a
n-1 r. N Y.E(f . IB* ) + n-1 r. N Y.E(f . IB* )
1E n1 1
n1 n
1£ nO 1
n1 n
= n-1 r.1ENn1Y.E(f
. IB* )
1
n1 n
+ 0
= n-1 r. N
-1
f
= n -1 r. N
-1
Y.{V
1E n1 1 n
r. N
•
.}
JE n1 nJ
Y.{V n}
1£ n1 1 n
= v -1
r. N Y.
n 1£ n1 1
= v n-1 L1ENY.g
1 n1.
V n > 1.
Therefore, by the Ra o -B1ackwe11 theorem, for any.convex loss L(a,b),
EL(y ,
n
y)
~ EL(y(v
)
n
, y).
In pnrticular,
- - -y) 2 > E(y(
_ y)2,
E(y
n
v )
n
V n > 1.
We write
f'
_n
Let
f
_n+1
g'
_n
~n+1
a
(fnl'··· ,fnN)' so that 1-N' f_n
= ~n
on vn and g •
-n
V n > 1.
+ ~n+1' ~n+l independent of :n and ~~~n+l
= (gnl, •.. ,gnN)'
= ~n
= n,
=
1, V n > O.
g'l
_n_ N = v n , n -> 1.
+ ~n+l' where the distribution of ~n+1 depends only
Thus, given Bn* , the distribution of f_n is generated
22
by the v :(N - v ): possible equally likely permutations, while
n
n
-1
v 1 can be a null vector with probability N v and a non-null
.n+
n
•
vector with probability (1 - N-lv ), there being N - v
n
•
n
equally
likely realizations: ~ = (vl' ••• ,v ) : vi = 0 for all but one i
N
. B* , f and v +1 (i.e., A and B* 1)
Thus, g1ven
n.n.n
n
n+
= O.
and g' v
.n
Therefore, {B * , n
are conditionally independent.
n
>
I} is a
transitive sufficient sequence.
Let B~
n
= B*1
* ••• ,B* for n
Bl,
n
BO*-measurable.
n
>
-
V ••• V B* be the smallest sigma field containing
n
1.
Then the events [.*
= t]
(or [.
= t])
Hence, if we are able to show that
E[YnIB~*] = E(YnIB:) = y(v)'
V
n
>
1,
n
O
then we would have for a B *-measurab1e stopping
time M
where
E[L(ym,
Y)IM = m]
~ E[L(y(v )'
Y)\M = m], V m
m
so that
EL(Ym'
Y)
~ EL(y(v )'
m
Y).
Towards this note that
= n -1 E.1£Nn1 Y.E(f
. IB0* )
1
n1 n
n
I 0*
= n-1 E.1£H Y.E
1 k=1E(uk.1 Bn ).
n1
11
> 1,
are
23
As
~
is independent of B*_ , j
k j
E(u
Note that v
Nkl
~
•••
B0* )
ki I n
~
1, we have
= E(u ki IB*k
v •.•
v B*n >'
•
as well as N are nondecreasing in k, and, hence,
kl
k
~
Nnl , V k
<
n.
•
Thus, given that i £ N , under
nl
B* v •.• V B* , i will also belong to N with conditional probability
kl
k
n
vk/v ' V k
n
~
n.
On the other hand, for every i £ Nkl ,
* = v -1 L N E (uk' I ••• )
k
1.£ kl
1.
I*
E (u 1.' B v ... V Bn)
k
k
= v k-1 E (L.1£Nkl ~.1 IB*k \I
*
••• vB)
n
= vk-1 E (1 I ••. ) =
Therefore,
for every i £ N l' k < n,
n
E(u . IB0* )
k1 n
=
-1
(vk/V n )V,t<
= v-I,
n
and
•
-1
= v L N Y.
n 1£ nl 1
TV n
> 1.
This characterizes the m1n1.mUm risk property of the sequential
y (\l ) for BO*-measurable M.
N
REFERENCES
Asok, C.
(1980), "A Note on the Comparison Between Simple Mean
and Mean Based on Distinct Units in Sampling with Replacement,"
The American
Statistician~ 34~
158.
l
24
Bahadur, R. R.
(1954), "Sufficiency and Statistical Decision
Functions," Annals of Mathematiaal
•
•
Basu, D.
Statistias~ 25~
423-462.
(1958), "On Sampling With and Without Replacement,"
Sankhya~
A~ 20~
Series
Chakrabarti, M. C.
287-294.
(1965), "Query and Solutions," Jou:rnal of
Indian Statistiaal
Assoaiation~ 3~
Chow, Y. S. and Robbins, H.
223-232.
(1965), "On the Asymptot ic Theory of
Fixed-Width Sequential Confidence Intervals for the Mean,1I
Annals of Mathematiaal
Statistias~
36~
457-462.
Ghosh, M. and Mukhopadhyay, N. (1979), IISequential Point Estimation
of the Mean When the Distribution Is Unspecified,1I
Communiaations In
Statistias~
Series
A~
8# 637-652.
Hajek, J. (1964), "Asymptotic Theory of Rejective Sampling with
•
Varying Probabilities from a Finite Population," Annals of
Mathematiaal
Statistias~ 35~
1491-1523.
Korwar, R. M. and Serfling, R. J. (1970), "On Averaging Over
Distinct Units in Sampling with Replacement," Annals of
Mathematiaal
Statistias~ 41~
Pathak, P. K. (1961),
liOn
Units in a Sample,"
2132-2134.
the Evaluation of Moments of Distinct
Sankhya~
Series
A~ 23~
415-420.
Raj, D. and Khamis, S. H. (1958), "Some Remarks on Sampling with
Replacement, II Annals of Mathematiaal
Statistias~
29~
550-557.
25
Robbins, H. (1959), "Sequential Estimation of the Mean of a
Normal Population," i>l'obability and Statistics (H. Cram~l'
Vo lurne) ~ VZf Grenander
edited~
Wiksell, Uppsala, Sweden.
235-245.
•
Almquist and
•
•
26
#
•
Comparisons of n** and E(V n*)
Table 1.
•
e
-1
n**
n*
E(v n *)
2.1
2.5
2.6
2.8
3.1
3.5
3.6
3.7
3.8
3.85
3
3
3
3
4
4
4
4
4
4
3
4
4
5
5
6
7
7
7
8
2.44
2.95
2.95
3.36
3.36
3.69
3.95
3.95
3.95
4.16
3.2
3.4
3.6
3.8
4.2
4.4
4.6
4.8
4
4
4
4
5
5
5
5
5
5
6
6
8
8
9
10
3.59
3.59
3.99
3.99
4.60
4.60
4.84
5.03
N
B
5
6