ON THE MINIMALITY OF A BOUNDEDLY COMPLETE
SUFFICIENT SUB FJELD
D. l3asu
\
Institute ot Statistics
M1meograph Series No. 401
August 1964
UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
ON THE MINIMALITY OF A EOUNDEDLY COMPLETE SUFFICIENT SUB-FIELD
by
D. Basu
University of North Carolina and Indian Statistical Institute
August 1964
This note, written for its pedagogical interest, attempts
at a simplification of a proof due to R. R. Bahadur (1957)
of the minimality of a boundedly complete sufficient
sub-field.
This research was supported by the Mathematics Division of the Air
Force Office of Scientific Research Grant No. 84-63.
Institute of Statistics
Mimeo Series No. 401
- 1
ON THE MINlMALITY OF A BOUNDEDLY COMPLETE SUFFICIENT SUB-FIELD
l
by
D. Basu
University of North Carolina and Indian Statistical Institute
SUMMARY
This note written for its pedagogical interest attempts at a simplifica)
I
tion of a proof due to R. R. Bahadur
complete sufficient
(1957)
of the minimality of a boundedly
sUb-fiel~t.
NOTATIONS AND DEFINITIONS
Let (t,Ji,~) be our probability structure. That is,1J is a family (P)
1J Of
of probability measures on a CT - field vr 1\ sub-sets of a sample space ~ •
Definition l:
The set N
E
Jt
peN)
Definition 2:
(a)
=0
for all P
(b)
The two
jJ.
'P - equivalent
An
E
-null if
-:9 •
The two sets A and B belonging to
SJ- equivalent
Definition 3:
?
is said to be
tit
J1
are said to be
1>
if their symmetric difference Abo B is
-null
-mble functions f and g are said to be
if the set {xl f (x) ~ g(x) liS
-mble function f is said to be
JIf I dP <co for all P
E
1>
1J
-null.
-integrable if
'P .
~iS research was supported by the Mathematics Division of the Air Force
Office of SCientific Research.
t
As usual we use the term sub-field to mean a SUb-CT-field.
- 2
Definition 4:
fi * of tit is said to be sufficient if corresponding
to each
P - integrable, v4- -mble f there exists an· Jt*-mble
f* such that, for all B e A * and P e ~
J f d P = J f* d P
A sub-field
B
B
The function f* is then called the conditional expectation of f
tA *
given
Definition
5:
and is determined upto a
The sUb-field
Jf o
. Jf
are those that are
Definition
6:
11 o
d P == 0
'P
satisfying the identity
for all
-equivalent to zero"
til
0
is
?
-equivalent to some member of every
alternative sufficient sub-field
jj *
P -integrable f
PEP
is said to be a minimal sufficient sub-field if each
member of
Now if
-equivalence"
is said to be boundedly complete, if the only
Jt.o-mble functions
bounded
y:>
be sufficient then for each
Jr *"
J1-mble and square
the conditional expection f* 1s also square
~ -integrable
and we have in addition
In other words,
J f2elF ~ J f2
*
elF
for all P
E
the sign of equalty holding for all P e »
f
if and only if f and f * are
rp - eqUivalent.
THE THEOREM
Theorem:
If
J.f 0
be a boundedly complete sufficient sub-field then
a minimal sufficient sub-field.
v40
is
3
rA * be
Proof: Let
any alternative sufficient sub-field and let A be an
J1 o.
arbitrary member of
We have to prove the existence of a set B
€
l:A *
'7')
J
-equivalent.
such that A and Bare
Let f be the indicator of A and let f* be the conditional expectation of
~ * and f *0 the conditional expectation of f * given
f given
Jt- o.
Since f is bounded we can, without any 106s of generality, assume that
both f* and f*o are bounded.
Now, from definition 4 we have, for each P
Jf
dP
= J f*
=
dP
,
P
€
Jf*o d P
Thus,
1(f - f*o) dP = 0 for all P
Ji 0 -mble
and f - f *0 is a bounded
completeness of
t.A o
€
l'
function.
From the bounded
it then follows that f and f....
~
are
~ -equivalent
and hence
for all
But we know that for all P
Sf2
dP
>
Therefore,
Sf2dP =
f and f* are
y
j
€
. . ?)
)"
f2 dP
*
j
~
f; dP
Jf2*0 dP
for all P
€
Y
• and hence
-equivalent.
Thus, the set
A
is
'P
= (x I f
(x) =1 )
-equ1valent to the set
B
and so B is the
= (x I f*
(x)
\11 *-mble
set we are searching after.
= l}
-
....
- 4
REFERENCES
Bahadur, R. R. (1957):
On Unbiased Estimates of Uniformly Minimum Variance,
Sankhya. Vol. 18, p. 211
I
Lehmann, E. L. and Scheffe, H. (1950):
Completeness, Similar Regions, and
Unbiased Estimation, Sankhya Vol. 10, p. 305
NORTH CAROLINA STATE
OF THE UNIVERSITY OF NORTH CAROLINA
AT RALEIGH
INSTITUTE OF STATISTICS
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standard or control. 1963.
345. Novick, M. R. A Bayesian indifference procedure.
1963.
346. Johnson, N. L. Cumulative slim control charts for thc folded normal distribution.
347. Potthoff, Richard F.
On testing for independcnce of unbiased coin tosses lumpcd in groups too small to usc
348. Novick, M. R. A Bayesian approach to the analysis of data for clinical trials.
~
J.
Fixed interval analysis and fractile analysis.
1963.
1963.
351. Potthoff, Richard F. On the Johnson-Neyman technique and some extensions thereof.
352. Smith, Walter L.
x'.
1963.
349. Sethuraman, J. Some limit distributions connected with fixed interval analysis.
350. Sethuraman,
1963.
1963.
On the elementary renewal theorem for non-identically distributed variables.
353. Naor, P. and Yadin, M. Queueing systems with a removable service stations.
1963.
1963.
354. Page, E. S. On Monte Carlo methods in congestion problems-I. Searching for an optimum in discrete situations. }'ebruary. 1963.
355. Page, E. S. On Monte Carlo methods in congestion problems-II. Simulation of queueing systems. February, 1963.
356. Page. E. S. Controlling the standard deviation by cusums and warning lines.
357. Page, E. S. A note on assignment problems. March, 1963.
358. Bose, R. C. Strongly regular graphs, partial geometries and partially balanced designs.
359. Bose, R. C. and J. N. Srivastava.
March, 1963.
On a bound useful in the theory of factorial designs and error conecting codes. 1963.
360. Mudholkar, G. S. Some contributions to the theory of univariate and multivariate statistical analysis.
361. Johnson, N. L. Quota Fulfilment in finite populations.
362. O'Fallon, Judith R.
Studies in sampling with probabilities depending on size.
363. Avi-Itthak, B. and P. Naor.
364. Brosh, I. and P. Naor.
1963.
1963.
1963.
Multi.purpose service stations in queueing problems.
On optimal disciplines in priority queueing.
1963.
1963.
365. Yniguez, A. D·B., R. J. Monroe, and T. D. ·Wallace. Somc estimation problems of a non-linear supply model implied by
a minimum decision rule. 1963.
Present value of a rcnewal process.
366. Dall'Aglio, G.
367. Kotz, S. and J. W. Adams.
368. Sethuraman,
J.
1963.
On a distribution of sum of identically distributed conelated gamma variables.
On the probability of large deviations of families of sample means.
1963.
1963.
369. Ray, S. N. Some sequential Bayes procedures for comparing two hinomial parameters when observations are taken in
pairs. 1963.
370. Shimi, I. N.
Inventory problems concerning compound pl'Oducts. 1963.
371. Heathcote, C. R.
queues. 1963.
.
A second renewal theorem for nonidentically clistrihuted variables and its application to the theory of
372. Portman, R. M., H. L. Lucas, R. C. Elston and B. G. Greenherg.
373. Bose, R. C. and J. N. Srivastava.
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Potthoff, R. F.
Estimation of time. agc, and cohort effects.
Analysis of irregular factorial fractions.
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Some Scheffe-Type tests for some Behrens-Hsher type regression problems.
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1963.