M A R G I N A L S U F F I C I E N C Y O F S T A T I S T I C S

V . N . S u d a k o v

We wish to d i s c u s s a p r o b l e m c l o s e l y a l l i e d with an i m p o r t a n t concep t in m a t h e m a t i c a l s t a t i s t i c s ,

n a m e l y tha t of su f f i c i ency . Le t a f a m i l y of p r o b a b i l i t y m e a s u r e s tPo, 0 e e ~ be g iven on a s p a c e with

a d i s t i n c t z - a l g e b r a (X, 0L). L e t us a s s u m e tha t a l l the m e a s u r e s of the g iven f a m i l y a r e a b s o l u t e l y

con t inuous with r e s p e c t to a c e r t a i n p r o b a b i l i t y m e a s u r e P . L e t ~ . ~c~) be a c e r t a i n m e a s u r a b l e

func t ion ( s t a t i s t i c ) de f ined on the s p a c e (X, b~). The s t a t i s t i c ~ i s s a i d to be su f f i c i en t fo r the f a m i -

ly P~ if fo r any s u b s e t A ~ 0~ the cond i t i ona l p r o b a b i l i t y P0 (A I S C~)) does not depend on the va lue of

the p a r a m e t e r O . F o r the c a s e in which (X, 0~,P) i s a L e b e s g u e s p a c e and we can speak of c o n d i -

t i ona l m e a s u r e s on e l e m e n t s of a m e a s u r a b l e p a r t i t i o n ~ g e n e r a t e d by the s t a t i s t i c ~ = ~(z~ the s u f -

f i c i e n c y of the s t a t i s t i c i m p l i e s tha t t h o s e cond i t i ona l m e a s u r e s a r e independen t of the p a r a m e t e r 0

on a l m o s t a l l e l e m e n t s of the p a r t i t i o n ~ . We i n t e r p r e t the e x p r e s s i o n " a l m o s t a l l " in the s e n s e of a

c a n o n i c a l m e a s u r e on the s e t of e l e m e n t s of ~5 , i . e . , a m e a s u r e on the f a c t o r s p a c e {X, ~ , P) /n~.

As i s a p p a r e n t f r o m the de f in i t ion , the s u f f i c i e n c y p r o p e r t y i s e s s e n t i a l l y a p r o p e r t y of a m e a s u r a b l e

p a r t i t i o n g e n e r a t e d by a s t a t i s t i c , so tha t we m a y of ten s p e a k of su f f i c i en t p a r t i t i o n s , r a t h e r than s u f f i -

c i en t s t a t i s t i c s , o r , f o r m a l l y in a m o r e g e n e r a l context , su f f i c i en t z - a l g e b r a s .

In add i t i on to the s t a t i s t i c ~ - ~{~ le t t h e r e be g iven on the s p a c e IX, ~t) s t a t i s t i c s ~ , ~ ) , . . . ,

~ - ~ l x ) �9 The fo l lowing p r o p e r t y of the s t a t i s t i c 5 c o n s t i t u t e s a g e n e r a l i z a t i o n of the su f f i c i ency

concep t : The c o n d i t i o n a l d i s t r i b u t i o n s of each of the s t a t i s t i c s ~ , , . . . , ~ s u b j e c t to the condi t ion of a

f i xed v a l u e of Sr do not depend on the p a r a m e t e r 0 e @ It i s c l e a r tha t if ~- ~(~ i s a su f f i c i en t

s t a t i s t i c fo r the f a m i l y {P~, 0e @t , then the cond i t i ona l d i s t r i b u t i o n of any o the r s t a t i s t i c qf~e) c e r -

t a i n ly does not e i t h e r depend on the p a r a m e t e r 0 e 0 o On the o t h e r hand, if the p a r t i t i o n g e n e r a t e d by

the s t a t i s t i c ~c~) i s c o a r s e r than the p a r t i t i o n g e n e r a t e d by ~ (~) , then the fac t tha t the cond i t i ona l

[ s u b j e c t to the cond i t ion ~ r ] d i s t r i b u t i o n of the s t a t i s t i c ~ does not depend on the p a r a m e t e r d o e s

not i m p l y tha t the s t a t i s t i c ~ r i s su f f i c i en t . Nor i s s u f f i c i e n c y of ~ t ~ i m p l i e d by the f ac t tha t the

c o n d i t i o n a l d i s t r i b u t i o n s of e ach of the s e v e r a l s t a t i s t i c s ~,r ~r do not depend on the p a r a m e t e r ,

even when we a s s u m e tha t the p r o d u c t of p a r t i t i o n s ~ r v ~ i s a p a r t i t i o n ~ of the s p a c e

(• 0 t , P) in to po in t s . F o l l o w i n g i s an e l e m e n t a r y e x a m p l e :

• =~=,, ~,, ~,, -,,},

T r a n s l a t e d f r o m Z a p i s k i Nauchnykh S e m i n a r o v L e n i n g r a d s k o g o O t d e l e n i y a M a t e m a t i c h e s k o g o

I n s t i t u t a im . V. A. S t ek lova A k a d e m i i Nauk SSSR, Vol. 29, pp. 92-101, 1972.

�9 1975 Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. IO0tL No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission o f the publisher. A copy o f this article is available from the publisher for $15. 00.

792

eo,ro,

For the case in which X = ~" and }j=) .... ,~.(=) are coordinate functionals [i.e., ~.(=)- =, ~ =o(=,,

. . , =~): ~ ~" ] the nondependenee of the conditional, distributions of the statistics },,..., }. subject to

the condition {(=) on the parameter 0r e is referred to as the property of marginal sufficiency of

the statistic } . The case most frequently met in statistics, of course, is when the distribution Pe

represents for each value of the parameter e a product of r~ measures (independent sample). Sev-

eral years ago the Indian statistician Vo S. Huzurbazar advanced the hypothesis that in the case of a

repeated sample.the marginal sufficiency of a statistic implies its sufficiency. The proof of this hy-

pothesis was announced in a preprinf published by Ghosh in 1968 [I], but my colleagues and I,-all

specialists in mathematical statistics, have run into considerable difficulties in attempting to recon-

struct the complete proof from Ghosh's preprint. We have not even been able to extricate a reasonably

c l e a r idea that might be of help in a r r i v i n g at the r equ i r ed proof . Below we se t for th the p roof of a

somewha t m o r e gene ra l p ropos i t ion .

When all p robab i l i ty m e a s u r e s of the fami ly {Pe, e e e } a r e abso lu te ly continuous with r e s p e c t

to a p a r t i c u l a r p robab i l i ty m e a s u r e , the suf f ic iency of a c e r t a i n s t a t i s t i c fo r any pa i r of d i s t r ibu t ion

funct ions {Pe,, P~,} in the fami ly {P0, ee @t impl ies , as we a r e well aware (see, e .g. , [2]}, the su f -

f i c i ency of that s t a t i s t i c with r e s p e c t to the en t i r e f ami ly in quest ion. We t h e r e f o r e confine our ana ly -

s i s to the case in which the se t ~ c o n s i s t s of two e l emen t s .

THEORE M 1. Let P and ~ be two mutua l ly abso lu te ly cont inuous Bore l m e a s u r e s in R ~

c o r r e s p o n d i n g to an independent sample . Let the s ta t i s t i c ~ ~(=~,. �9 : , =~) be m a r g i n a l l y s tdf ic ient ,

foe., fo r a l m o s t all va lues of ~ with r e f e r e n c e to the d i s t r ibu t ion of } let the condi t ions (C~, Pv) /~ , =

=(C~, ( 2 1/%, ,~ ~, , �9 �9 j ~ hold, where ~, a r e coord ina te pa r t i t ions and P~ and Q~ a r e cond i -

t ional p robab i l i ty m e a s u r e s on an e l emen t C~ e R ~ of the pa r t i t i on ~ . Then 5 is a suff ic ient s t a t -

i s t ic fo r the pa i r of d i s t r ibu t ion P and ~ .

In the case r~ = Z , and only in tha t case , it suf f ices to r equ i r e in p lace of independence of the

sample sa t i s f ac t ion of the condit ion cLO./gP= o~(~,) % (~.).

F o r a r b i t r a r y n. the sample independence condit ion can be r e p l a c e d by the following: F o r a l -

m o s t e v e r y n u m b e r ~ with r e f e r e n c e to the d i s t r ibu t ion ~ the re a r e funct ions P~(=,),,.., P~(=,) and

n u m b e r s 5~ and B v s u c h t h a t 0-5~p,~I=,)-B~ r , ~ = ~ , . . . , n , and ~ - V = p , ( % ) . . . p ~ l = ~ ) o

We f i r s t p rove some a n c i l l a r y p ropos i t ions :

PROPOSITION 1o Let P and (~ be two mutua l ly absolu te ly cont inuous p robab i l i ty m e a s u r e s

on the space ( ~ , 0~), and let ~ be a m e a s u r a b l e par t i t ion fo r which t he re ex is t s y s t e m s {P,} and

{0.~} of condi t ional p robab i l i ty m e a s u r e s ( C is an e l emen t of the pa r t i t ion ~ ). If d~O./d.P = q ( ~ ) ,

then on a lmos t e v e r y e l emen t C ~ ~ the m e a s u r e s P, and a~ a r e mutua l ly abso lu te ly .cont inuous ,

and d~O.,/dP~ = ~(C)~{co) [ cocO, and ~(C) is the value on the element C of the density function

].

Proof. It suffices to show that the system of measures ~ on elements of ~ having the follow-

ing densities with respect to the measures Pc :

793

~P, a ( a / ~ ) a P '

is in fac t a sys tem of cond i t iona l measu res f o r the measure Q under the p a r t i t i o n

~(~ O.(A~. =!~(c~)d,P = ! =

�9 We obtain

PROPOSITION 2. Let U and Y be f in i te nonnega t ive Bore l m e a s u r e s in R ~ such that for

some

~ , 0

2. V { z - [ = . . . . . . =..)~R'~: 5 =, ~ &}: 0

3. (R~ U)/~, =(R-', V)/~,,, K= I . . . . , r~

Then U = V = 0.

In the case n = Z the conc lus ion a l so holds without the four th condi t ion; for ~ �9 Z the condi t ion

that the f i r s t abso lu te m o m e n t s ex i s t cannot be d i s c a r d e d .

Proof . Let us a s s u m e that U ~ 0 and V ~ 0 . By the equa l i ty of the m a r g i n a l d i s t r i b u t i o n s the

f i r s t abso lu te m o m e n t s a l so ex i s t for the m e a s u r e V . F o r the s a m e r e a s o n the point &- (m, . . . , m~),

t ! =" dU, I, ~ ac t s as where m. = [ffl. K . . . . , , the b a r i c e n t e r of both the m e a s u r e U , and the m e a s u r e

Y �9 It fol lows f r o m condi t ions 1 and 2, on the o ther hand, tha t the b a r i c e n t e r s of the m e a s u r e s U

and V a r e e s s e n t i a l l y s i tua ted on d i f fe ren t s ides of the hype rp l ane def ined by the equa t ion Z =, = & �9

The r e s u l t i n g con t r ad i c t i on p r o v e s the p ropos i t ion .

In the case a = z for any nonnega t ive Bore l m e a s u r e s U and V sa t i s fy ing condi t ions 1 and 2

i t i s p o s s i b l e to p ick a func t ion } (=) such that a t r a n s f o r m a t i o n F of the p lane R ~ tak ing the point

~=, ~) into the point (~(=), ~-~(~-~)) and thus mapp ing each of the h a l f - s p a c e s g iven by the equa t ions

in to i t se l f , we t r a n s f o r m the m e a s u r e s U and V into m e a s u r e s UF ' ' and VF "~ having f i r s t m o m e n t s .

I n a s m u c h as coord ina te p a r t i t i o n s a r e i n v a r i a n t unde r the t r a n s f o r m a t i o n F , t h e equal i ty of the m a r -

g ina l d i s t r i b u t i o n s holds for the t r a n s f o r m e d m e a s u r e s as well , i .e . , a l l the condi t ions 1 through 4 a r e

s t i l l sa t i s f i ed , so tha t the m e a s u r e s UF -~ and VF -~ , a long with U and V t h e m s e l v e s , a r e , a cco rd ing

to the proof, z e ro -va lued �9

794

We show, f ina l ly , t ha t f o r a > ~ the s a t i s f a c t i o n of cond i t i ons 1-3 does not g u a r a n t e e the e q u a l i -

ty U = V -- 0 . We c o n s t r u c t in R' two f in i t e p o s i t i v e m e a s u r e s U and V , which a r e c o n c e n t r a t e d on

d i f f e r e n t s i d e s of the h y p e r p l a n e d e s c r i b e d by the equa t ion ~ + :% �9 % o 0 , and have i d e n t i c a l m a r g i n a l

d i s t r i b u t i o n s . We adopt a s U a p u r e a t o m i c m e a s u r e (of t o t a l m a s s ~/~ ) c o n c e n t r a t e d on the h y p e r -

p l a n e = ~ § :

t h r e e m a s s e s of ~ each a t p o i n t s with c o o r d i n a t e s (~, ~,- t / , (~, - 1, 1), (- ~, I, ~ ;

t h r e e m a s s e s of ~ each a t p o i n t s wi th c o o r d i n a t e s (-~, -1, 3), (-~, 3 , -1) , (3, -~, -~) ;

. . . . . . . . . . . . . . . . . . . �9 , �9 ,

t h r e e m a s s e s of 2, " c " * ~ ' each a t p o i n t s with c o o r d i n a t e s

1-Z"~), (t-z *~, 1-..z "~, Z' -I)

. . . . . . . . , . , * , . . . . , , o . ,

H e r e K=Z~ 3, . . . . F o r the m e a s u r e V we adop t the i m a g e U q-~ of the m e a s u r e [J u n d e r the

m a p p i n g q: R ~ ~ R 3 , ~ ( ~ , ~ , % ) ' ( - =~, - ~ , - ~5) ~ I t i s r e a d i l y v e r i f i e d tha t the m a r g i n a l d i s t r i b u -

t i o n s fo r both the m e a s u r e U and the m e a s u r e V co inc ide and a r e c o n s t r u c t e d as fo l lows : m a s s e s

of t/~ a t p o i n t s -1 and +~ �9 m a s s e s o f 2. - ~ + ~ a t p o i n t s l - z " and z ~ - I K~z, 3,

P r o o f of the T h e o r e m . We f i r s t e x a m i n e the s i m p l e r c a s e in which the u n i v a r i a t e d i s t r i b u t i o n s

~, and ~ , K = I , . . , ~ , w h o s e p r o d u c t s a r e , r e s p e c t i v e l y , m e a s u r e s P and (~ a r e such tha t ~ ~ 5

and ~a~* ~ B . Le t the m a p p i n g q: R ~ - - * R" t ake the po in t ( ~ , . . . , %) into the po in t ( ~ q~(%~, . . . ,

The mapp ing ~ t a k e s the m e a s u r e s P and (1 in to m e a s u r e s P = Pq-~ f~ %(=~)), w h e r e %c=~)= ~, .

and ~=(2~-', while the measures Pc and a~ map into certain measures ~ = P~ ~-~ and (~= a, ~-~,

, ~ c~(P/~) ( P r o p o s i t i o n 1). Thus , q-~(~, . . . ,~ c o n s i s t s of the w h e r e a~/d~P~ = K~C)e:~p ~=, z , and ~(C)= - ~ ( f f / ~ -

po in t s (:c~, ~ ~'~) fo r which ~ r~(~ ' , )= ~ so tha t fo r t h e m ~--~-~ = ~ ( C ) % ( = : ) . . . 9~ ( ~ ) = ~(C) e z ~ " ~ = "' ' ripe

~(C) e ~ =" , and u n d e r a mapp ing ~ c o n s t a n t a long the " l e v e l l i n e s " of the d e n s i t y ~ the d e n s i t y of

i m a g e s of m e a s u r e s at a c e r t a i n po in t i s equa l to the d e n s i t y of t h e i r i n v e r s e i m a g e s a t any i n v e r s e

i m a g e of tha t poin t .

The condi t ion , p r e s u m e d h e r e to be s a t i s f i e d , t ha t the o n e - d i m e n s i o n a l d e n s i t y func t ions have

u p p e r and l o w e r bounds , i m p l i e s the s a m e cond i t ion of t w o - s i d e d b o u n d e d n e s s fo r the l o g a r i t h m s of

t h o s e d e n s i t i e s , i . e . , c o o r d i n a t e i m a g e s (a l l up to s u b s e t s of m e a s u r e z e r o ) . In o t h e r w o r d s , the mea,~

s u r e s P and fi , a long with the m e a s u r e s Pc and ~r , a r e c o n c e n t r a t e d in a bounded p a r t of R ~

s p a c e and t h e r e f o r e have a f i r s t m o m e n t .

Now l e t

w h e r e U and V a r e d i s j u n c t i v e nonnega t ive m e a s u r e s ~ C l e a r l y , the m e a s u r e s U and Y s a t i s f y

the cond i t i ons of P r o p o s i t i o n 2. Thus , cond i t i ons 1 and 2 a r e i m p l i e d by the f ac t tha t

795

so that the a l t e rna t ing m e a s u r e

i.e., f o r

and is nonposi t ive for

'~a~ = K(O)e=p 7_. = ,

Q~- Pc is nonnegat ive w h e r e v e r

K(C)e,~p~ x. > "1,

y__.=, > - t~ K(C),

~ - t r ~ K ( C ) .

In condi t ions 1 and 2, t h e r e f o r e , it is r e q u i r e d to put e=-s

Condit ion 3 s igni f ies the p r e s u m p t i o n of m a r g i n a l s u f f i c i e n c y , and condit ion 4 follows, as we have

shown, f r o m the hypothes i s of boundedness of the un ivar ia te dens i ty funct ions . Consequent ly ,

v = v o o , i.e.,

i .e . , in fact , the pa r t i t i on g is suff ic ient fo r the m e a s u r e pa i r P and CL .

In the case ~ = z P r o p o s i t i o n 2 holds without the r e q u i r e m e n t of ex i s tence of the momen t s , so

tha t fo r ~ ~ m a r g i n a l suf f ic iency impl i e s suf f ic iency as long as the dens i ty ~ can be f a c t o r i z e d d. P~, on e v e r y e l emen t C~ of the pa r t i t i on ~ , i .e . , can be r e p r e s e n t e d in the f o r m P,~(~J P~ ( ~ ) . In the

case of a r b i t r a r y ~ our l ine of r ea son ing should r e m a i n una l t e red as long as we m e r e l y r equ i r e two-

s ided boundedness of the funct ions p~(z~), . . . . p~)(z~) involved in the fac to r i za t ion .

We tu rn now to the ge ne ra l ca se and prove that the m e a s u r e s U and V st i l l have a f i r s t absolute

moment . Now for the f i r s t t ime we mus t r e l y heavi ly on the fac t tha t the d i s t r ibu t ions P and & c o r -

r e spond to a r epea ted sample (so fa r we have r e l i ed only on the weaker condit ion of f ac to r i za t ion of

the condi t ional dens i t ies ) .

F i r s t , let the s t a t i s t i c ~ be such that the fol lowing condit ion holds f o r a ce r t a in L~o :

C lea r ly ,

Also, let

P[C~o) - O.

d.o.o P{C) d,P~: CL(C) (~'(=')'" " %'(~')"

796

We obtain

whence

and

d~--'~-" : g e=p s~

~ ( ~ ) = { R e=p s-l) -~ Ucdm).

We now consider the partit ion of the space (R, ~ U) into hyperplanes

then we let {U, } be a family of conditional distributions (probability measures) on the hyperplanes

{Nst and let c~(~s) be the corresponding factor measure on the set of such hyperplanes (i.e., the

values of the variable s on the set I- ~r~, ~ ) ] , so that

U = T U~a(c~s) - f , ~ -R

and

Clearly, the following inequality holds:

whereupon

~ ,e~cPm, t = dQ t 1

(we make use here of the assumption of an independent sample), i.e.,

Next we introduce the symmet r ized measure U obtained f rom U by the averaging of all

of the measure U under all possible permutat ions of the coordinates of ~ space:

2_ uf', ~ .

n! .~mages

797

w h e r e G,,, i s the g roup of p e r m u t a t i o n s of ~ c o o r d i n a t e s . I t i s obvious tha t the a v e r a g e s ~ , of the

c o n d i t i o n a l m e a s u r e s U, co inc ide with the cond i t i ona l m e a s u r e s fo r the m e a s u r e ~ . F o r the a v e r -

age m e a s u r e ~ a s we l l a s fo r the m e a s u r e U the fo l lowing cond i t ion ho lds :

~, 4~R Reap s-1 '4~.~ Re~p~-1

We now show tha t the m e a s u r e ~ h a s f i r s t a b s o l u t e m o m e n t s , wi th i t s b a r i c e n t e r s i t u a t e d a t

the po in t &~ = (s, s , . . . , ~) �9 Thus , i t fo l lows f r o m (1), in p a r t i c u l a r , tha t

fo r a l m o s t a l l

and

s [with r e s p e c t to m e a s u r e a (c~s) ], so tha t

w h e r e the i n t e g r a l s a r e e v a l u a t e d o v e r r e s p e c t i v e s u b s e t s of ~ s p a c e of the f o r m {~: 0-~ % - r and

( = : s - =~ ~ ~ t �9 But the m e a s u r e ~ can be r e p r e s e n t e d a s the l i m i t of i t s r e s t r i c t i o n s to s p h e r e s

of r a d i u s - v ~ with c e n t e r a t the po in t Bs " E v e r y such r e s t r i c t i o n ~ i s a s y m m e t r i c ( i nva r i an t

u n d e r the group q,~ ) m e a s u r e , so tha t fo r i t

P a s s i n g to the l i m i t on

S C=,- s) U~ (r I I=,-s l ~ (u,~. $ - o o

in t h i s equat ion , we obta in

~ $

S - v O

and the l a t t e r equa t ion d e s c r i b e s the b a r i c e n t e r of ~s �9

Next we p r o v e tha t a l e f t - s i d e d a b s o l u t e m o m e n t e x i s t s f o r the m e a s u r e

4=R =~=-oo -~.R =~-s

oo oo oo

-t~R Re:eps-1 =e~p~%

i t s e l f :

[by v i r t u e of (1)].

798

If the marginal distr ibutions of the measures Po and (2~ coincide, the marginal distr ibutions of

U and Y also coincide, so that the existence of lef t-s ided firs~ moments of the measure U (as ' i s

implied by the proven existence of the lef t-s ided f i rs t z, -moment of D ) and of r ight-s ided moments

of the measure V implies the existence of all f i r s t absolute moments and, hence, the validity of

Proposi t ion 2, i.e., in the given case PC = (2~ .

But ff P(Cv)= 0 for a lmost all ~ , khen as before, when Proposi t ion 1 was used~ we obtain for a l -

most all

where

measure P relative to its distribution with respec t to measure Q at the point ~ .

and

R~ is the value of the density function for the distribution p of the statist ic } with respec t to

Inasmuch as

the following condition holds for /z =almost all values of ~ :

o o

since

whence we infer, exactly as before, the existence of left-sided moments for U~ and thus use P ropos i -

tion 2 to prove the equality Pc = 0~c , Joe., we deduce the required sufficiency of the stat ist ic ~ o

The foregoing example of two distributions in ~ ' concentrated on different sides of the hype r -

plane s = 9 and having identical marginal distributions enables us to show that the condition

r~

does not guarantee the sufficiency of a marginal ly sufficient statist ic ~ . Indeed, if U and Y are

measures in ~ with identical marginal distributions as in the example analyzed above, we investigate

in ~5 probabili ty measures P and {2. defined by the equations

P'= aU+ 6V, 0.= a.eU § ~-~V,

799

in which

It is eas i ly ver i f ied that

L~ t ~ e o-=-~ e--- ~ , ~ ~ - ~ - ~ .

O~C~ d P = e~cp ~,. e~p mz. e~p m~

and the marg ina l d is t r ibut ions of the m e a s u r e s P and CL coincide, i .e. , that an unequivocably nonguf-

f icient s ta t i s t ic identical ly equal to a constant is margina l ly sufficient.

1~

2~

L I T E R A T U R E C I T E D

J~ K. Ghosh, Marginal Sufficiency, Indian Statist. Inst. Techo Rep. Math. Statist. , No. 11 (1968).

J . Ro Bar ra , Notions Fondamenta les de Statistique Mathematique, Dunod, P a r i s (1970).

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