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ON MAXIM\L AND Mm!MAL SUB-FIELDS CF CERTAIN TYPES
by
D. Basu
University of North Carolina
and
Indian Statistical Institute
Inatitute of Statistics Mimeo Series No. 422
January 1965
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This research was supported by the National Science
Foundation Grant No. GP 16-60
DEPARTloENr OF STATISTICS
IDT.IVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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ON MAXIMAL ArID Imrn-1AL SUB-FIELDS OF CERTAIN T1"PES
by
D. Basu
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University of North Carolina and Indian Statistical Institute
Let
(!, ~ ~) be a civen statistical mdel.
't'le concern ourselves i'rith
particular families of sub-fields of!" and, for each sucll i'amily we ask ourselves i'tl1ether the family ias maximal and minimal elements iri til respect to the
For example t~le family (~)
natural partial order of inclusion relation.
sub-fields tl18.t are sufficient for
from tlleoretical statisticians.
off ~).
shoi'm [1] tl18.t (~)
-..L,
has received a ereat deal of attention
!d~)
Clearly,
and is tile !naximum element
It is known [4] tlmt, in general, (~) has no m.i.niInl element.
ever, if i'Te assume that
an;)~ S., €
~
E.
is dominated by a
(S) and any A€ S ,there exists
-<>
vthere!:i
E
0
for all
HOiT-
a-finite meaS1.U'e then it can be
~,
has an essentially mininn.un element
P(AA B)
of
1. e., given
B€ S., such that
-..L,
P€
E. ,
stands for the operation of symmetric difference.
Let
Ji be a fixed sub-field of !
sub-fields tl18.t are independent of Ji, i.
P(BC) == P(B)P(C)
for all
and let (.£)
e.,
B € Ji,
be the familiy of aJ.l
for any SUCJ.l
C € £, and
£
it is true that
P € ~.
TIle minimum element of (.£) is the trivial sUb-field consisting of the nullset and the whole space.
Tileorem 1:
He Imve the following
For any sub-field .£ that is independent of &, there exists
at least one r:axilnal. subfield
C*
such tl)8.t
Q. c Q.*•
The proof of Theorem 1 is C;iven in the next section•
For the next problem, let us suppose that the family
P
is indexed by
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t"l0 parameters
e
and cp, i. e. ,
Consider a.ll sub-fields ~}
tllat are generated by statistics Whose probability
distributions do not involve the paramter cp, i.e., each
such tllat,for every D€ -,
D Po.,cp (D) is a function of
e
Q
is a sUb-field
only.
Tile trivial sub-field is again the minimum ele:r..ent of ( Q)j but, doea
this fanrl.ly have ma.x:imal elenents?
lie have
Tl1eore."ll 2:
such tba.t the restriction of
Given any sub-field D
-
Pe
,cp
to D
does not involve cp, tl1ere e7..ists a maxin:al such sub-field Q* tl1B.t contains
The above Theorem is an immediate generalization of Tlleorem 1 :in [2].
In the next section ue eive tIle proof of the above t1'l0 theorems.
In
the final section we comr:Jent on some fUrther problems of tIle same kind.
2.
We need the following
well-~~mm
Lenr.1a 1 (Zorn' s Lemma):
ever~/ linearly ordered
elel:Jent
x
set
that
SUC:"l
Proofs of Theorems 1 and 2
lemmas.
If for a partially ordered set it is true that
sub-set has an upper (lower) bound, then given any
of tIle set t11ere exists a maximal (minimal) eleL-ent
x is
~ (Ojreater) than
*
JC
in the
*
x.
[The terms that are underlined are defined in terms of the partial order
reJ..a.tion].
LenE8
2 (Extension of Measures):
Given a measure
I.l.
defined on a field F
of sets there exists one and only one extension I.l.* of!t to t}le Borel-extens:i.on
F* of F.
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Corollary:
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meo.sUl~es
1.1 and v
aeree on a field
necessarily asree on the Borel-extension
Lemma
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If the two
3:
If the family (~)
of sub-fields of
A
F
.J!-
E.
01'
sets
the~'
of F.
be linearly ordered "TitIl
respect to the inclusion rclation then
-;a
=
U B
-a
is a field of sub-sets.
He omit t11e proofs of the lemmas.
HOlT let
all sets
B be a givcn sub-field of A and let
E€ A
such that
is independent of
E
~
E
bc the class of
i. e.
P(EB) ~ P(E)P(B) for all B€ ~ , E€ §. and P€
It is
eas~r
to cllec]{ that E
fl1X-'cJlcr tllat
In case
t:.1Cll
E
contains t11e null-set anc1. tIle
space and
is closed :Lor co:;"[)lementation and cOlUlto.blc disjoint unions.
E
is a
cr-field tl1cre is nothing to prove in Tjlcorem 1, as
tile ua.ximal sub-field for lTl:d.cl1 we are searchinG.
usuaJ.l~r not a
~ti.lole
!: .
Hmrever,
E
E
is
is
cr -field (see exaqple 1).
Let {.9a} be a
famil~T
of sUIJ-fields in
E
anel be lincarly· orderea. 1Titll
re:::pect to the inclusion rclation and let
£o=U£a.
Hon, from Lemma 3,
every
nember of
~
£0
is a field of sub-sets of A and, since
C is indeTlcndent of
-0
J:;'
Consider the two meaSLJ~~es
A € A.
B.
-
Choose and. fL"{
20
P€.p.
and :'1ence, from the
corolla.ry to Lemma. 2, they B.u7ee over the Borel-extension
B a.nd
B€ B and
!J
p(AB) and P(A)P(B) defined for all sets
These two measures Bi.:;ree over the field
Remembering tl18.t
Co c
P
£0*
were arbitrary oembers of
respectively, we noW' ha.ve
3
of
C.
-0
Band P
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P(A)P(B) for all AE C* ,
p(AB) -
*
C
Tlms,
--0
includes every
Ler;]":1B. I are satisfied and
l!
and is independent of
~
-....
~lence
and
B.
-
PE P •
TIle conditions of
the proof of Theoren 1 is complete.
He noi" turn our attention to Theorem 2.
Let F
be tlle class of all
-
sets F € A such that
P e (F)
,cp
AGain, it is easy to
is a function of
checl~
that
e
only.
£: contains the mill.-set and the whole space
and is closed for conq>lementatian and countable disjoint unions.
2 ire sllall see that
Let
CIa) be
!:.
is usually not a sub-field of
a famiJ.:l of sub-fields in
£: and
In eX8.IiIple
!.
be linearly ordered ,·Titll
respect to the inclusion relation and let
As before
of
l20
is a field of sub-sets of X.
He define the measure
[Since
~
c £:, the measure
From Lemma. 2, for eaC}l
fi.o*
Let
tJ.;
be tIle Borel-extension
~.
is 1.U1ique.
must ar,-ree on
Q
e
on
~
as the restriction of P e q>
e on ~ must be independent of
e € e , the extension Q*e of Qe
Q
on
~.
~
to
q>.]
from
From the corollar,{ to Lemma. 2, the two j:1easures
*e
Q
and P e cp
*
~.
In other i'lOrds, the restriction of
dependent of cp.
Also,
P e,cp
to tIle sub-field
~ includes every ~.
4
*
~
is in-
The conditions of Lemma 1
are satisfied and hence tJ.le proof of Theorem 2 is corrIplete•
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,2.
E~~!IJ?les
and Comments.
The following two e:2tr1l?les demonstrate tbat the
element is
mBJ~]al
l'lsuall':l not unique.
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Exagple 1:
Let
X consist of the four points
a, b, c and cl
and let
E.
con-
sist of just one probabillt:: distribution namely t"ile uniforr.l distribution over
the four points.
£.1
~
Consider tile tllI'ee sub-fields
and
.ge
each consist-
inC of four sub-sets of X:
B
and
Here,
consists of
!'
(a, b) and their conwlenents,
£.1 ••••••••••• !., (a,c)
2e •• • • • • • • • • • !, (a, d)
eacil of £.1 and ge is
maxinal such sub-field.
..................... ,
••••••••••••••••••••
independent of
£.1
Incidentally,
and
!
•
and eacil of them is a
£e
are also independent of
each other.
E:::a.qple 2:
!
Let
consist of the five points
a, b, c, el, and
e
and let tile
probabilit.y distribution over the five points be
Points:
a
Probs:
1-0
i'li1ere
0 <
The f'a.m:i.1jr
e< 1
!.
and
c
b
d
e
0 < cp <~. •
of all sets irllose probability does not involve q>
consists
of 12 sets, that is
b
(a), (b,d) , (b,e J,
(c,el)
, (c,e)
and their complements.
Note til&t F
does nat constitute a sub-field.
There are t'uo different
nn:ti.mal sub-fields in !., naj:nel:y,
i:t
consistinC; of !, { a), (b, d), (c, e )
and tlleir conw1ements.
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and
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(a), (b,e), (c,d)
and their complements.
T1leorems 1 and 2 only establish the existence of maximal sub-fields
in E
respectivel~r.
and F
It would be of sane interest to develop general
!
metllods for proving the ma.x:ima.lity of certain given sub-fields of
and E,.
One such method, with very limited application, is given in Theorem 7 of [2].
Consider the problem where we have
"J.'x , ... , x
2
n
on a real random variable
n
independent observations
x
with cumula.ti ve distribution
function of the form
- co < cp < co,
"There the fUnction F
is Immm and
e
0
< e <co •
and cp are the so-called scale and
location parameters.
P
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b
consisting of
If
y
stand for the vector-valued statistic
tllen tIle distribution of
y
does not involve the loca.tion parameter
y
a. maximal such statistic?
sub-field generated by
of T1leorem 21
y
t11en is it true that
Bc A
T1le sub-field
mapping
C
--y
C
-"
is maximal in the sense
Some further problems
i3 s;'.i.d to be sufficient for tile sub-field
if for every C€ C there exists a
B-measurable function
X into the real line such that
P(BC) ==
1
f(x
C) dP( x)
for all
B
P€ P
and B€ B.
6
Is
be the
The author does not expect the answer to be 'yes' for all F.
4.
£ c!
In the language of sub-fields, if
cp.
f(x; C)
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In other words,
B is sufficient for
C if •for
every
.,
-
C~ C
)
there
1!
exists a choice for the conditional probability (function) of C given
tJ18.t serves for all P€ P.
Now, for a fixed
fiels
of
C such that
-
(C)
1!
~
let us enquire about the family (Q) of all sub-
is sufficient for
Clear!:Jr, the minimum elellEnt
is t·he trivial sub-field consisting of only the null-set and the
Whole space.
.
Do there exist maximal elements in {C} ?
Let Q. be the class of all sets
for
C.
G€ A such tl18.t
B is sufficient
G in the sense mentioned above, n&llEly, for every G€ G there exists
a B-measurable
f(x;G) such that
P(GB)
E
f
f(x;G)dP( x)
for all
P€ P
B
and B€
'l'l1e class
in that
1!.
Q. is similar to the classes
E anf F
considered before
G contains the null-set and the Whole space and is closed for
com'ple)~nta.tion and
a sub-field.
countable disjoint unions.
The rest of the
argument~
As before,
g, is usually not
in Theorems J. and 2 will apply if "re
could prove a result of the following type:
"If B is sufficient for each member of a field Q of sets in
1!
then
is sufficient for the Borel extension
C* of
~
f."
The above statement does not seem to be true in the Generality stated
above.
In the particular case "fuere
posed above has a definite answer.
for
B is the trivial sub-field, the question
For, in this case,
1!
can be sufficient
G if and only if
p(G) is the
S8J.i1e
for all
P€ P ,
and tlrerefore, Theorem 2, or rather a particular case of it, n&llEly, Theorem 1
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in [2] applies.
Fraser in [3] introduced the notation of partial sufficiency in tile
follo""Ting manner:
If
P
-
= (P e,cp ), e € e ,
cpt
be a family of probability measures in-
Q ,
dexed by the two independent parameters
Hill be called e-sufficient for A
i)
the restriction of F e,cp
e and
cp, then a sub-field
(or simply e-sufficient) if
to ~ does not depend on cp [i. e.,
.!! is a sub-field of the tJPe considered in Theorem 2.],
and.
ii)
given any A€ A , there exists a choice (of the conditional
probability (function) of A e;iven
eac:l
G € ® there exists a
Fe
~
~
that does not depend on 8, i.e., for
f e (x;A) such that
.!!-measurable function
(AB)5!f (x;A)dP e
B o
,CP
ft
for all
B€B
and all
(O,cp).
Under what conditions does a e-sufficient sub-field exist?
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.!! c A
Does there
e:dst an essentially minir.lI.uu such sub-field?
AJ;
a final problem on the existence of minima.l sub-fields consider
the following:
Given two sub-fields
sub-field that contains both
Now, for a fixed
fields
Q.
c!
~
~
and
£J
9€
!! V £
stand for tIE smallest
and Q..
.!! c!, let us consider the family
(Q.)
of all sub-
such that
!.
~v£=
EveIj-r
let
(Q) may be called a complement of .!!.
its nm::inn.un element.
tile allSirer to be yes.
Does
(Q.)
The
fa.mil~l
have minimal elements?
(Q.) has
~
The author expects
It is easy to construct exa.m;ples 1-There there are
several minimal complements to
~.
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References
1.
Baluuiur, R.R.
Sl~ficiency
(1954),
and Statistical decision functions,
Ann. Math. Statist., 25, 425-462.
2. Basu, D. (1959),
~le
family of ancillary statistics,
3. Fraser, D.A.S. (1956):
Ann. Math. Statist.
4.
21, 247-256.
Sufficient statistics lr.ttil nu.isance para.r:'lters,
27, 838-842.
Pitcher, T.S., (1957): Sets of measures not admittinG necessaIJr and
sufficient statistics or sUb-fields,
P
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S~1ya,
9
Ann.
~Bth.
Statist., 28, 267-263.