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CoNFlIlENCE INTERVALS FOR THE l'lEAH
OF A FJlUTE POPUlATION
V. M. Joshi
Institute of Statistics Mimeo Series ITo. 475
May 1966
This research was supported by the Mathematics
Division of the Air Force Office of Scientific
Research Contract No. AF-AFoSR-76o-65.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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1.
Introduction •
TIle admissibility of estimates of the population total inth the Squared Error
as the loss function were considered by the author
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II) and a certain estimate
was shovm to be always admissible whatever be the sampling design, in the entire
class of all estimates.
This estimate is equivalent to using the sample mean
as estimate of the population mean.
Here we consider the allied question of
admissibility of the confidence intervals for the population mean based on the
sample mean and the sample standard deviation, which are also commonly used in
practice.
These confidence intervals are here shown to be also always admissible
whatever be the sampling design.
The result is proved for a wider class of sets
of symmetrical confidence intervals of which the above mentioned confidence
intervals are a particUlar case.
2.
Notation and Definitions.
~~;
The population u consists of N units u l ' u 2 ' ••• ,
= 1,2,
associated the variate value Xi' i
I
(1965,
a point in the Euclidean N-space
•.. , N; X
= (Xl'
irith the unit u i is
x2 ' •.. ,
XN)
denotes
R:N; a sample s means any subset of u; S denotes
the set of all possible samples s; on s is defined a function p such that
p(s) 2: 0 for all s and
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p(s)
=1
S€S
The samples s are drawn at random irith probability p(s) and the sampling design
d
= (c,
p).
Then we define
DEFINITION 2.1 An estimate e(s,x) is a real function e defined on S x ~
which depends on x through only those Xi for which u i
€
s.
The above definitions of sampling design and estimate are wide enough to
cover all sampling procedures and classes of estimates; for a brief account we
refer to Godambe and Joshi
(1965,
I, Section
5).
We next define admissibility of a set of confidence intervals for the population mean,
I
~
(1)
i
=
--I
N
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Xi
i=l
For a given sampling design d, we denote by
S,
ing of all those samples s for ~mich pes) >
o.
the subset of S, S C S, consist-
Now let el(s, x), e (s, x) be
2
two estimates (Definition 2.1) such that e (s,x) ~ e (s,x) for all x € ~ and
1
all s
€
2
S; then (e l (s,x), e2 (s,x») denotes the set of confidence intervals
e , e , x C S, con1 2
sisting of all those samples s for which (2) holds, and for any alternative set
For every x
€
R_, let
-~
Se ,. e , x denote the subset of
2
l
S, S
of confidence intervals (ei(s,x), e2(s,x») let S,
e , e f , x,
of
S, S,
e ,
l
,
e ,
2
1
x
C
denote the subset
2
S, consisting of all those samples s for which
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We now define
DEFINITION 2.2 The set of confidence intervals (el(s,x), e (s,x») for the
2
population mean is admissible if there exists no other set of confidence inter-
2
vals {ei(s,x), e (s,x») such that
(i)
e (s,x) - ei(s,x) ~ e 2 (s,x) - el(s,x) for all
x € ~ and for all s € S
2
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pes) 2:
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for all x
€
l1i
the. strict inequality in (ii) holding for at least one x
€~.
The sums
in (ii) are obviously the inclusion probabilities for the confidence intervals.
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and
(ii)
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We also define a weaker version of admissibility by,
DEFINITION 2.3
The set of confidence intervals (el(s,x), e (s,x)} for the
2
population mean is weakly admissible if there exists no other set of confidence
2
intervals (ei(s,x), e (s,x)} such that
(i)
e (s,x) - ei(s,x) ~ e2 (s,x) - el(s,x)
2
X
and
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(ii)
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pes) >
and all s
€ ~
for all
E
S,
,
pes)
SES
e ,e ,x
l 2
SES e I ,e I ,x
l 2
for almost all (Lebesgue measure) x
holding on a non null subset of
€ ~,
the strict inequality in (ii)
~.
To distinguish the admissibility as defined in Definition 2.2 from weak admissibility we shall refer to the former as strict admissibility.
Ie
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above definitions of admissibility of confidence intervals are based
on the definition for infinite frequency functions with en unknown parameter
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formulated by Godambe (1961) and subsequently slightly modified by the author
(1966)
3.
Main Result.
For a sample s the sample mean
.1
(4)
Xs
= n(s)
\.
L,
iEs
where i
E S
X
s
is given by
x.
~
is written shortly for
U.€S
~
and is so written hereafter and n(s)
denotes the sample size i.e. the number of units in the sample s.
The sample standard deviation is
VI
(s,x)
=[
nCs)
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iES
3
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The usual confidence intervals based on the sample standard deviation are then
= Xs
= Xs
el{s,x)
(6)
e {s,x)
2
- k v'{s,x)
+ k v'{s,x)
where k is a positive constant) usually
=2
or 3.
However we shall prove the re-
sult for the more general class of confidence intervals given by
= Xs
= Xs
el{s,x)
e {s,x)
2
- v{a,x)
+ v{s,x)
where v{s,x) is any arbitrary non-negative estimate (definition 2.1).
As a
first step towards proving strict admissibility we prove
THEOREM
'.1
in (7) is
The set of confidence intervals (el{s,x), e 2 {s,x»
Suppose the set is not weakly admissible.
Then by definition 2.3,
there exists an alternative set of confidence intervals (ei{s,x), e {s,x»
2
,
satisfying
e {s,x) - ei{s,x) ~ e 2 {s,x) - el{s,x)
2
(8)
x
and
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(9)
p{s)
for aJmost all x
Note:
€
~
Ill,
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€
III and all
s
for all
€
S,
p{s)
the strict inequality in (9) holding on a non-null set.
Throughout the rest of this paper the measure considered will be the Lebesgue
measure on
Ill.
So also for any k demensional subspace of
~,
the measure con-
sidered will be the Lebesgue measure for the k dimensional subspace.
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weakly admissible.
PROOF
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When the
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measure considered is for a k-dimensional subspace, it will be indicated by
(~)
TheBe points will not be repeated each time.
Now put
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.i(s,x) + e2(s,x)
- v(s,x)
2
(10) and
+ v(s,x)
2
Then
e~(s,x) = e'1 (s,x) +
(11)
S ei(s,x)
-He (s,x) - ei(s,x) - 2v(s,x)]
2
by (8)
and
2
e (s,x)
(12)
= e2(s,x)
2
- ·Me (s,x) - ei(s,x) - 2v(s,x)]
2: e2(s,x) by (8),
so that denoting by S"
e , e " ,x for every x
2
l
e" (s x) <
(13)
l'
x._
€
R_, the subset of'S, on which
-N
< e" (s x)
- -"N -
2
'
we have from (11), (12) and (3)
(14)
(14) combined with (9) now gives
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(15)
p(S)2:
for almost all x
set of
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pes)
,
the strict inequality in (15) holding on a non-null sub-
~.
We now take the expectations of both sides of (15) w.r.t. a prior distri-
5
.
bution
such that all the xi' i=1,2, ••• ,N are distributed identically.
~ on~,
We denote as usual by P[D], the probability assigned under the distribution
to any subset D of
~;
~
further the set D will be denoted by specifying the re-
lation satisfied by the co-ordinates of the points x
€
D.
Then on taking ex-
pectations of both sides of (15) we have
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(16)
pes)
P[e~(s,x):s ~:s e~(s,x)]2:
pes) P[e l (s,x):s
~ :s e2 (s,x)]
s€S
s€S
Now putting in
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(10)
ei(s,x) + eg(s,x)
= .......---------
e(s,x)
2
and using
(7), (16) is reduced to
(18)
L
pes) P[e{s,x) - v{s,x) oS
~
oS e{s,x) + v{s,x)]
S€S
>
~
L
pes) P[x - v(s,x) <
s
-
~_
< xS
-~ -
+ v(s,x)]
s€s
As the x., i
~
= 1,
2, ••. , N, are by assumption distributed identically, the value
of each probability occurring in (18), remains unaltered if in each of the functions
~(s,x), v{s,x) and xs' we replace the xi for i
by xl' ~, ••. , xn{s)'
€
s ~aken in some particular order
Let by this substitution, the functions e(s,x), and
v(s,x) be respectively transformed into functions f (x) and w (x), which depend
s
s
on x only through the co-ordinates xl' x2 ' •.• , xn{s)'
formed into
n{s)
xn(s)
=
1
n(s)
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i=l
Xs is obviously trans-
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-,
Then (18) is transformed into
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\' p(s)P[fs(x) - ws (x) -< ~< f (x) + Ws (x)]
-~ - s
(20)
~
Now putting
1
XN-n(s)
(
(21) -i
(x)
g
s
= N-n(s)
f
= --:;;s
(x)
_
1 -
and noting that
~ = p(~~ Xn(s)
~-n(s)
+ (N-n(s»
N
N
(20) is seen to reduce to
\' pes) peg (x) - u (x) < ~- () < g (x) + U (x)]
s
s
- -~-n s - s
s
(22)
~
-
SES
->
\' pes) P[X'n ( s ) - u s (x) -< X-_
() < x ( ) + u s (x)]
-~-n s - n s
~
SES
So far the only assumption regarding the prior distribution
the x., i
~
~
on
~
is that all
= 1, 2, •.. , N are distributed identically. We now make the further
assumption that all the x., i
~
= 1,2,
a Normal distribution with mean
e and
•.. , N are distributed idependently with
variance ~2 so that
7
x.,... N(e,cr).
= 1,
i,
1.
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2, ••• , N
and
~-n(s) -
cr)
N(e,
For given values of the
..J H-n(s)
l~ n(s) co-ordinates xl' X2, •.. , xn(s) let
and
Then for the given values of xl' x ' ..• , xn(s) the conditional probability
2
(23)
peg (x)-u (x) < 5[_ () < g (x)+u (x)lx ,
s
s
- -~-n s - s
s
l
= IN-n(s)
. h1f cr
J
t +u (x)
s
e
2cr
z
2
dz
t -u (x)
s
= ¢(Itsl)
s
-
2
.lli::E.W.L...
2
s
say where ¢(Ixl) is strictly decreasing as Ixl increases.
Here we have written ~(It s I) briefly for ¢( It s I, u s (x», were u s (x) may depend
on; (). By ¢(It I) being strictly decreasing is meant that ¢(It I, u (x»
n s
. s
s
s
strictly decreases when It I alone is increased, the values of u (x) being
s
s
held fixed.
Similarly the conditional probability
=¢
(IT I)
S
Also the freiUency function of the variables xl' x2 ' .•• , xn(s) may be written
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in the form
(25)
-
L(Xl , "', xn(s) 1xn(s)
) e
n( s' 2
- ~(x ( )-e)
2(l
n s
Hence transposing the r.h.s. of (22) to its l.h.s. and using (23), (24) and (25),
we have, writing x for short for xl'
"2' ... , xn(s) and dx for short for
dxl , dx2 , ••• , dxn(s)
(26)
\' J
~ pes)
-:~)
L(XIXn(s»
(Xn(s) _8)2
[¢(Igs(x)-el-~(Ixn(s)-el)]dx
e
20
S€s
We nmf integrate the l.h. s. of (26) w.r. t.
e from
- a to + a.
As the integrand
is non-negative we may by Fubini's theorem interchange the order of integration
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The l.h.s. in (27) is non-decreasing as a increases.
converges to a limit or
~ co.
\'
L
~~,
it either
In either case we have
~
(28)
So as a
(»dxJde
p(s)fL(xli
'J
ns
-
n(s) -
2
2 [xn(s)-e]
e 2cr
[¢(Ig (x)-el)-¢(Ii -el)] >0
s
s-
-00
In the l.h.s. of (28), the integral w.r.t.
e
vanishes at any point x=(xl , x ' .•• , xn(s)
2
We shall now show that at any P9int at which gs (x)fX:n ( s )'
at which gs (x) = Xn ( s ).
this integral is < O. For denoting the integrand by I, the integral w.r.t.
in (28) may be expressed as
L
Xn(s)+gs(x)
2
I dO +
9
e
Now suppose that gs (x) > Xn (s).
Then in the second integral in (29), transform
the variable of integration by putting
eso that
Also
(30)
e-
Ss (x)
= Xn ( s ) x
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01
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+g
e goes from _ 10 to nCs) sex)
1
2
~n(s)
= gs(x)
-
el ,
so that the second integral becomes
_ n(~) (ss(x)-e )2
l
dBl e 20"
[¢( IXn(s)-Gll)-¢( Ig s (x)-9l )]
J
Writing in (30),
e
for
el , and combining
with the first integral in (29), (29)
reduces to
gs (x)+xnCs)
J
2
de~( Igs(x)-el)-¢( IXn(s)-el) Ie
~
2
~ ~(s)-e)
ill!l
20"
- e
2
Since by assumption gs(x) > Xn(s)' we have in the range of integration in (31),
Ig s (x) - 81> Ixn ( s ) - el
so that since ¢(Ixl) strictly decreases aslxlincreases the first factor in the
integrand of (31) is always < 0, While the second factor is > 0 and hence (31)
and so (29) < O. By a similar argument (29) is shown to be < 0 if at the point
if
x'Ags(X) < Xn(s). As (29) is the same as the integral w.r.t. e in the l.h.s.
of (27), it follows from (27) that the set of points x
= (~,
x2 ' ••. xn(s»
which
is for every B
depend on x
€
Consequently since g s (x) and xn (s )
S a null (~n ( s » set.
= (Xl'
x2' ••• , ~) only through Xl' x2 ' ••• , xn(s)' the set of
10
2
- n~ (gs(x)-~
- n
at
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points x € ~, at which (32) holds is a null (~N) set.
definition of gs(x) in (21) that
fs(x) = Xn(s) a.e. in~, for all s € S
(33)
and since f s (x) is the transformed form of e(s,x) in (18), we have from (33)
e(s,x)
= x S a.e.
in~,
fer all s € S
Hence the strict inequality in (15) and consequently in (9), cannot hold on a
non-null set.
4.
The assertion in Theorem 3.1 is thus proved.
Extention of Theorem 3.1.
We have now to prove that the weak admissibility of the confidence inter-
vals proved in Theorem 3.1 implies their strict admissibility.
A similar result
in connection with the admissibility of a certain estimate was proved by the
author (1965, III,sections"and 5) and the argument here runs closely parallel.
We consider the hyperplanes in B:N obtained by assigning fixed values to some
k of the variates.
Let ~-k be the hyperplane in which say the last k variates
JeN-l+t' t = 1, 2, ••• , k have fixed values ON-'k+t
respectively.
Let Sk be
the subset of S, consisting of all those samples which contains each of the last
k units ~-k+t' t=1,2, ••• ,k, i.e. s
Sk' if, and only if, un _k+t €S for all
t = 1, 2, ••• , k and s € S. We denote by Sk' Se ,e ,x the intersection of the
l 2
set Sk with the. set Se ,e ,x on which (2) holds; the intersections of Sk with
l 2
the sets S, , on which (3) holds, and S II II
on which (13) holds, are
e1 ,e2 ,x
e~,e2'x
_
denoted similarly. Now suppose that for X€~_k' and s € Sk' estimates ei(s,x),
e~(s,x)
€
exist such that,
pes) >
pes)
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It follows from the
11
holds for almost all ( ~-k) x
h(S,x) = e(s,x) -
€
xs f
~-k and that further e(s,x) being as in (17)
0 for at least one s
for a non-null (~-k) subset of points x
supposition leads to a contradiction.
€
Sk
€
~-k. We shall show that this
As before
(35) implies .
(36)
for almost all (~-k) x
2
€
~~-k. Now in (35) and (36), the estimates ei(s,x),
2
e (s,x), and hence ei(s,x), e (s,x) and e(s,x) are defined only for samples
s
€
-Sk and points x
x
€
~ and to samples s
Let
~~k
€
a
~-k.
We now extend their definitions to other points
ft
Sk as follows.
be the hyperplane C
~
given by
~-k+t = a'N_k+t'
We establish a 1-1 correspondence between the points x
by putting
(37)
x; = xr +a, r= 1, 2,
••• , N-k
the constant a being so fixed, that denoting by
x's = _"""il~_
n(s)
(38)
N
-
1
\'
XN=T Lxi
i=l
we have, for all X
€
(-k and all S
€
Sk'
€
a
t
= 1,2,
"N-k and x'
€
a'
..• , k.
'tN-k
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This can always be done, for putting
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a'r
(40)
Ci =
~
N
I
ar
r=N-k+l
we have
x's
=~
[k
n\sJ
-
XN
1
= T[k a' + N ~r
x.!.
-n
- x's = x._
- xs - k(~
- ...L)(a'
- ex - a)
-~
n\sJ
N
a'
+ n(s)
xs - k a
+ (n(s)-k) a]
\.
and
- k a + (N-k)a ]
so that
so that (39) is satisfied if
(41)
= at
a
-a
We now introduce a new estimate
and s
€
~(s,x) defined as follows; for x
€
~-k
Sk'
(42)
and for x
(43)
Since by
t
*v(s,x)
= v(s,x)
a'
-
*v(s,x)"
= v(s,x)
€
~-k ' s € Sk
(42) and (43) ~(s,x) is defined for every hyperplane ~~k' it is
defined for all x
€ ~,
condidence inter\~ls
for all s
€
Sk'
In place of the original set of
(el(s,x), e2 (s,x)} in (7) we conaider a new set
(et(s,x), e~(s,x)} where
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*v(s,x)
+ *v(s,x)
-.
ef(s,x) = xs -
(44)
e~(s,x)
= xs
We denote by Se*,e*,x, the subset of S , for each x
l
€
2
R_, for which
-N
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et(s,x) ~ ~ ~ e~(s,x) holds.
(45)
We now extend the definitions of ~(s,x), e1(s,x) and e~(s,x) to other hyperplanes
a'
~-k
by putting
x' - x s
e(s,x') = e(s,x) + s
e~(s,x')=
(46)
e"(s,x')=
2
-e(s,x')- *~(s,x')
-e(s,x')+ *v(s,x')
-
Since for x € ~-k' ef(s,x) and e~(s,x) coincide with el(s,x) and e2 (s,x), for
a
every x € ~-k
_I
and hence from (36)
(47)
for a1Jnost all (~-k) x
(48)
€
~-k and further from the observation below (35)
h(s,x) = e(s,x) -
Xs f
0 for at least one s
€
Sk '
a'
for a non-null (~-k) subset of ~-k'
Now
~:k'
(47) and (48) are easily seen to hold for every other hyperplane
For, for any point x'
E
~:k'
by
(44), e!(s,x') = x~
14
- ~(s,x')
= xs-;(s,x)+(xs-xs )
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by
iN = ~ + (X~ - xs)
(46), and (43), e (s,x') = ei(s~'x)
1
,',,"
(39),
..
and by
+ (x~
- xs>'
e~(s,x') = e (s,x) + (x~
- is)
,.
2
a'
so that for every s8Jll1)1e s € -Sk' at every point x' € ~-k'
et(s,x') ~ ~ ~ e~(s,x')
holds, if aild only if, at tne corresponding point x € ~-k defined by (37)
el(s,x) ~ ~ ~ e2 (s,x)
and similarly
e"(s x') < x:._ < e (s x')
l'
- -"'N - 2 '
holds, if and only if
1
e (s,x)
S ~ S e2 (s,x)
holds.
Hence
(Note remark below (46»
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From (49) it follows that (47) holds for almost all (~) x
R and since
N
a' contains a non-null (~-k) subset on ,,1h1ch (48) holds,
every hyperplane ~-k
£
the subset of ~ for which (48) holds is non-null (~).
Now consider samples s
f..
Sk' Le. s
2
€
the values of e(s,x), ei(s,x), e (s,x), and
for s-
€
(S.Sk) and all x
€
(S - Sk)'
~(s,x)
We have so far not defined
for s
€
(S - Sk)'
We now put,
~
e(s,x) = x s
*v (s,x) = v(s,x)
e"(s,x)= e(s,x) - *
v(s,x)
1·
v(s,x)
e (s,x)= e(s,x) + *
2
eI(s,x) and e~(s,x) being given by (44).
15
Then for s
€
(S - Sk)' and all
X€
~
I
I
-.
= e!(s,x)
e~(s,x)
= e~{s,x)
2
e (s,x)
so that
(50)
combining (471 (48) and (50)
I
(51)
>
pes)
for almost all x
E ~,
(52)
= e(s,x) -
h(S,x)
for at least one s
E
-
and
xs f
0
5, for a non-null (~) subset of ~.
But Theorem 3.1 applies to the set of confidence intervals (e!{s,x),
Hence by
~.
(34), the set of points on which (52) holds must be a null
(~)
e~{s,x»).
set of
Thus our Driginal assumption below (35) that
h(S,x)
= e(s,x) - Xs f 0 for at least one s
subset of ~-k must itself be false.
THEOREM 4.1
E
Sk for a non-null (~-k)
We thus have
If estimates ei{s,x), e {s,x) exist, such that (35) holds for
2
almost all (~-k) x E ~-k' then with e{s,x) as in (17), h{S,x)
vanishes for all s
E
Sk' for almost all
16
(~-k) x
€
~-k .
= e{s,x)
- xs '
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5. Strict Admissibility•
We now come to the final part of the argument.
The argument is closely
similar to that of Theorem 5.1 proved by the author (1965, III), but as the
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proof was given there for a fixed sample size design, while here we have a varying size design, we shall again indicate the main steps in the proof.
Let EC ~ be the set of all points x €~, for which h(S,x)
for at least
is empty.
OlWS
€
= e(s,x)-xsfO
By (::54), E is a null set, and we have to show that it
S.
Suppose it is not empty, then there exists at least one point
r
x = a = (a , a , ••• , ~) and one sample So € S such that h(so,a) = ho o.
2
l
Without loss of generality we may suppose the sample s to consist of the first
o
m units u l ' ~, ••• , um. Consider the (N-m) dimensional hyperplane P;-m defined
by
a
1 = 1, 2, ... , m.
X € :B
if and only if, x.~ = a.,
-m
~
For every x
x
N
a
P
N-m
€
1
s
=m-
o
m
I
a. = a
~
say
0
1=1
a 0 +h0
e(s , x) =
0
v(s , x) = v(s 0 , a)
0
and
ma
0
~=
Hence for x
x
€
so
N
+
~
H
I
x.~
i=m+l
a
P
N-m
v(so,x)
S~ SX
So
+ v(so'x) holds if, and only if,
N
&'o (1 - .J!L)
- v(s 0 ,a) <
N
i i=m+l
I x.
~
and similarly
17
< a0 (1 - ~)
+ v(s 0 ,a)
1'1
-
I
I
-.
e~(s,x) = e(s,x) - v(s,x) ~ ~ ~ e(s,x) + v(s,x) = e (s,x)
2
holds, if and only if
N
ao (1
(55)
l
- ....!!L) - v(s ,a) + h < N
N
0
0
\'
L
-
xi <
i=m+l
-
a (1
0
a
We now determine a subset ~-m of pN_m such that for x
but (55) does not, as follows:
(56)
If h
>
o
0
X€d3'
,
~-m
m ) + v(s ,a) + h
N
.0
0
-
€
~-m' (54) holds
if and only if,
,
N
ao(l- : ) + v(so,a) <
I
~
Xi
~ ao(l
i=m+l
and if h <
o
-
: ) + v(so,a) + ho
0
ao(l -
: ) - v(so,a) + ho
~ ~
N
I
Xi <a8(1 - : ) - v(so,a)
i=m+l
Since ho f. 0, ~-m is always determined and has infinite measure (~-m)'
Since we are considering strict admissibility now, we assume that the
alternative confidence intervals (ei(s,x) e (s,x»
2
are such that as required
by condition (ii) of Definition 2.2,
I
holds for all x
(58)
I
holds for all x
pes)
€
~
pes)
€ ~,
>
and not merely for almost all x
€~.
Then by (14)
>
the strict inequality holding for at least one x
Now by (56) for every point x
€
Q~
~-m
,
18
€
R "
N
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and
S
o
E
s
f.
0
s-e
l
,e ,x
2
Se h ,e ., ,x
l 2
and hence at every such point, there must be at least one other sample s
for which h(S,x)
= e(s,x) -
xs f
0
S,
as otherwise at this point the r.h.s. of
(58) inll exceed its l.h.s.
Let ~~
F~~ be the subset of all the points x E pa
for Which h(S,x) r~ 0
N~
for at least one S E S. Obviously
Q~
'l'l-m
C R~
~-m
and since Q~ is of infinite measure (u.. ), R~ is of infinite measure (IJ...
'l'l-m
'~-m
~-m
'~-m
We now partition the set R~ into (not necessarily) disjoint subsets indexed
T
) •
~-m
by the samples s
E
S.
Let for a specified sample s, h~'S be the subset consist~-m
ing of all those points x
x
€
h~'s
~-m
€
R~
-I'j-m
for which h(s,x)
if, and only if, x
E
R~
~-m
f
0
i.e.
and h(S,x) ~ O.
r
Then from the definition of ~-m if follows that
(60)
a
-
EiI-m -
U- h~'S
~-m
SES
Since R~
has infinite measure (u..
~-m
. l\J-m ) there must be at least one non-null (IJ.._
. !\J-m )
set in the r.h.s. of
(60).
Now there are two possibilities:
(A)
For every non-null (~.
) set h~'S
in the r.h.s. of
'~-m
~-m
(B)
contains all the units u l ' u2 ' ••• , um with in addition some other units;
There exists at least one non-null (~N ) set h~'S in the r.h.s. of (60)
-m
~-m
for which the sample s, contains only some k, (0 ~ k < m) out of the first
m units •
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E
19
(60), the sample s~s
r 0,
We shall first consider that(B)holds and there exist one or more non-null sets
L.~,s in the r.h.s. of (60) for which the sample s does not contain all the first
1'l-m
m units; if there are more than one such set we select one of them arbitrarily.
Let
a s
LN:r:
be the
set selected and suppose the sample sl is of sample size
k, (0
that it contains
~
k < m) out of the first m units, the remaining
units being from the last (N-m) units um+l ' um+2 ' ... , ~~.
r!" \. Since a' E PNa -m ,we have
a' € -:I'l-m
~
(~-k)
Then take any point
( a l , a2 , .•• , am' am+l , am
"
a '=
+2 .•.}
, ~
(61)
(N-ml ) dimensional hyperplane
Then for the point a' we define as in (53) a
a'
P
by
N -~
(62)
x
i
i
for i
€
sl'
i<m
-- ati
for i
€
sl'
i>m
= a
Next putting
a. +
~
a'
a'
we define as in (56) a subset 0:,1 C P
L~-ml
(63)
and, if h
x
€
,
C~_m
~
= h(s 1 ,
1
if, and only if,
N-~
by
a'
XE
P
N-m
l
a') > 0
I
m
1
1
+ h 1<
--x < a (1 - ---)
- Nil
N
ils l
20
and
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ell
I:
I,
I:
I
I:
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-.
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.e
Next in (63), assign to all co-ordinates xi for i
fi
Sl and i > m fixed values
equal to the corresponding co-ordinates of the point aI, i.e. for i satisfYing
Thus in the inequalities in (63), we replace the middle
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term by
1
(64)
a!~ +
N
and since sl contains k out of the first units, there are (m-k) values of i for
!
which i
subset
and i < m.
sl'
a
Hence the inequalities define a (m-k) dimensional
l
~-k
.
al
al
Clearly ~-k is of infinite measure (Il _k ) in the hyperplane P - , i.e.
m
mk
aI
at
c P
"'m.-k
m-k
0-
the hyperplane p
(65)
aI
m-
xi = a i
x. =
~
a~
J.
k being defined by
for i
E
i <m
sl'
i > m.
for
at
a
a ~
The hyperplane P k is wholly orthogonal to P
and hence the set 1&-:' ~ wi th the
N-m
m-n-m
set
~~k
defined for each a tEL:t~::?- ' by substituting (64) for the middle term
in(63), determine a set
P;-k being the hyperplane defined by xi =ai for each i, such that i
Combining (63), (65) and (66), the explicit definition of the set
€
sl' i ~ m.
n:_k is given
below
... ,
X E
Since
r;~; is
~~k
for some a I
if, and only if,
€
r.;~:l
of positive measure
(~-m~ and
21
for each a I E
r;~rt
the set
~~k
has infinite measure (Pm- k ), the set D:_khas infinite measure (~-k).
o
I
I
Here
< m.
~ k
a
How let EN-k
F0
which h{S,x)
Il.~
-N-k
(66)
C
C
a
P - be the set consisting of all those points x
N k
for at least one s
a
E
N-k
S.
€
a
P - for
N k
€
Then
and hence the set ~-k is of infinite measure (~-k).
We now again partition
the set ~-k into subsets by
Ft
= u L..':'
s
~-k
-n-k
tt€S
(67)
the subsets ~~: being defined for each specified s
x
€
r.;~:, if and only if x
€
S by
€
~-k and h{s,x) F 0
Again at least one of the subsets in the r.h.s. of (68) must be non-null (~~-k).
If there is only one non-null (~-k) set r.;~:, such that s
each of the k units ui with i • sl' i
does not include
~ j
k units u i given by i
€
sl' i < m.
< k) out of the
Then again proceeding as from (63) to (67)
we reach a set ~_j of infinite measure (~_j) such that for every x
h{S,x)
=0
for at least one s
€
S.
Clearly the process can end only when we (A') either reach a set
such that ~ has infinite measure (~) and for every x
some s
(B')
€
-S;
~_j'
€
~r' h{S,x)
€
~ C ~
F0
for
or
we reach a hyperplane
P;_j
defined by some j, (o <
j
S m)
out of the first
m variates having fixed values equal to the corresponding co-ordinates of the
point a, i.e. x
r
a set ~_j
h{S,x)
f'
C
= ar ,
for r
=i l ,
i , ••• , i j Where i l , i , ••• , i
2
j
2
~
m and
P:_ j such that ~~_j has infinite measure (~_j) and for every x
0 for some s
€
S, and further such that for
22
any sample s
€
S, which
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m, we select it; if there are more
a s2
than one such subset, we select one of them arbitrarily. Let ~:k be the
~
subset selected and suppose the sample s2 contains j, (o
-.
€
~_j'
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-.
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does not include each of the
j
... , u . J.,
units u.
~
~l'
•e
h(s,x)
=0
for almost all
Here we note that the possibility (A) below (60) is included in the case
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(B'), the corresponding value of
being = m.
j
How (A') is not possible because E
=E
N
3.1.
(B') also leads to a contradiction.
consisting of all those samples s
Then for every s
€
(s -
(~_j)
forms a null
€
and E is a null
set by Theorem
For let Sj be the subset of S, S c Sj
S, which include each of the units u. , u. , ..• , u ..
,
~l
Sj)' the set of points x
set.
(~)
€
P;_j for which h(S,x)
~2
F0
Consequently introducing intersection sets as previously
in (35), we have,
I
(68)
I
=
pes)
pes)
S€(S-S ). S
j
for almost all(~_j) x
I
pes)
P;_j"
€
e l ,e2 ,x
(68) combined with (58) gives then,
>
for almost all (~_j)' x
€
P;_j
But then"by Theorem 4.1 already proven)the set of points x
which h(S,x)
f
€
~;_j for
0 must be a null (~_j) set, while according to (B') ~_j is
of infinite measure (~_j).
Thus neither (A') nor (B') is possible.
that h(s,a)
F0
for some s
E
-S,
Hence no point a
E exists such
and thus the set E is empty.
Consequently the sign of strict inequality in
does not hold for any point x
€
E~.
(58) and consequently in (57)
We have thus proved
TheDrem 5.1 The set of confidence intervals (el(s,x), e 2 (s,x)} in (7) is
23
~j
II
II
strictly admissible.
6.
e.i
Remarks.
The theorem proved here gives an optimum property of the sample mean as an
estimate of the population mean.
The previous result of the author (1965, II)
also defines another optimum property of admissibility of the sample mean and
it is of interest to compare these results.
the squared Error as the loss function.
The previous result is based on taking
Now the squared error is one of several
possible loss functions, and there may exist other reasonable loss functions,
for which the sample mean is not always an admissible estimate.
From this aspect,
the property proved here, which is independent of any loss function
m~
be said
to have a deeper significance.
The previous result was not subject to the restriction of measurability.
The present obviously holds in the class of confidence intervals determined by
measurable estimates.
But as discussed previously by the author (1966, IV) the
restriction of measurability is of no practical significance.
Acknowledgment
This investigation was
sugges~ed
by a discussion with Professor Godambe regarding
the Empirical Bayes character of confidence intervals based on the sample mean as
proved by Godambe
(1966).
I have also to thank Mrs. Kay Herring for doing the
typing work neatly and speadily.
References
(1) Godambe V. P. (1961) Bayesian shortest confidence intervals and admissibility.
Int. Inst. de stat.
(2)
Godambe V. P.
rd
33-- session Paris.
(1966) Empirical Bayes shortest confidence intervals for the
mean of a finite population.
(3)
Annals of Math. stat. No 3, Abstract.
Godambe V. P. and Joshi V. H. (1965) Admissibility and Bayes estimation in
sampling finite populations.
24
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Ann. Math. Statist.
36, 1707-1722
II
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(4)
Joshi V. M. Admissibility and Bayes Estimation in sampling finite populations.
II; Ann. Math. Statist. 36 1723-1729.
(5)
Joshi V.
M. Admissibility and Bayes Estimation in sampling finite populations.
III Ann. Math. Statist. 36, 1730-1742.
(6)
Joshi
v.
M.
Admissibility and Bayes Estimation in sampling finite populations
IV (submitted for publication to Ann. Math. Statist.).
(7)
Joshi
v.
M. Admissibility of confidence intervals, Ann. Math. Statist. (to be
published in June, 1966 issue).
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