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Problems Relating to the Existence of Ma.x:i.mal and
Minimal Elements in Some Families of Statistics (Sub-fields)
by
D. Basu
University of North Carolina
and
Indian Statistical Institute
Institute of Statistics Mimeo Series No. 435
June
1965
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This research was supported by the Air Force Office of
Scientific Research Contract No. AF-AFOSn-76o-65.
DEPARTMENT OF STATISTICS
UNIVEf{SrrY OF NcmTH CMOLINA
Chapel Hill, N. C.
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Problems Relating to the Existence of Maximal and
Minimal Elements in Some Families of Statistics (Sub-fields)
by
D. Basu
(University of North Carolina
and
Indian Statistical Institute)
1.
SUMMARY:
In statistical theory one comes across various families of statistics
(sub-fields).
For each such family, it is of some interest to ask oneself
as to whether the family has maximal and/or minimal elements.
The author
proves here the existence of such elements in a number of cases and leaves
the question unsolved in a number of other cases.
A number of problems of an
allied nature are also discussed.
2.
INTRODUCTION:
Let (x. , G , p) be a given probability structure (or statistical
IJX>del).
A statistic is a measurable transformation of (x. ,G) to some other
measurable space.
Each such statistic induces, in a natural manner, a sub-
field* of G and is, indeed, identifiable
wi.th
the induced sub-field.
Between sub-fiel.ds of a there exists the following natural partial
order of inclusion relationship.
Definition 1:
The sub-field ell is said to be larger than the sub-field G2
if every member of G2 is also a member of ale
*abbreviation for sub-a-field
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A slightly weaker version of the above partiaJ. order is the following:
Definition 2:
sub-field
a2
The sub-field
al
is said to be essentiaJ.ly larger than the
if every nember of 02 is P-equivalent to some member of
al
•
[Two measurable sets A and B are said to be P-equ1vaJ.ent if their
symmetric difference MB is P-null for each PEP.]
Given a family :J of sub-field (statistics), one naturally enquires
as to whether CJ has a largest and/or least element in the sense of definition
1.
In the absence of such alements in 3, one may enquire about the possible
existence of maximal and/or minill'Bl elements.
[An elenent of
ao
of 3 is a
JIBXimaJ. (miniDBJ.) element of 3, if, there exists no other element
such that
al
ao.]
is larger (Smaller) than
al
in 3
In the absence of ~ l (minimal)
elements in 3, one may look for elements that are essentially largest (least)
or are essentiaJ.ly maximal (minimBJ.) in thesenae of the weaker partial order
of definition 2.
The particular case where 3 is the family of all sufficient sub-fields
The largest element of :J is clearly the
has received considerable attention.
total sub-field
a
itself.
I f P be a dominated family of measures then it is
well-known that 3 has an essentially least element in terms of the weaker
partial order of definition 2.
minill'B1 elements.
In general, 3 does not have even essentially
If, howeVer, an essentiaJ.ly minimal elenent exists then
it must be essentially unique and, thus, be the essentiaJ.ly least elemnt
of 3 (see Cor. 3 to Theorem 4 in [3]).
In [1] the author considered the family 3 of ancillary sub-fields.
[A sub-field
ao
is said to be ancillary if the restriction of the class P
of probability measures to
.
measure. ]
a0
shrinks the class down to a single probability
The least ancillary sub-field is clearly the trivial sub-field
2
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consisting of only the eIIg?ty set
ma.x:iJm.l. elements in the
[1].
In
fami~
41 and the whole space 'I.
of ancillary sub-fields was deIOOnstrated in
In general there exist a multiplicity of rna.xima.l ancillary sub-fields.
sections:3 to 6 '\ore list four problems that are similar to the problem
of ancillary sub-fields.
In section
7 we develop a general method to deIOOn-
strate the existence of maximaJ. elements in these four cases.
'We
Existence of
discuss some related questions.
In section 8
In section 9 we list a number of other
problems.
3. (B-independent sub-fields (31 )
Let lB be a fixed sub-field.
A sub-field C is said to be lB-independent
(independent of lB) if
P(BC) == P(B)P(C)
for all B
lB, C
€
Let 3
1
€
C and P
be the
least element of 3
1
fa.mi~
€
P •
of all lB-independent sub-fields.
is the trivial sub-field.
Even in very simple situationsJ
3 1 has no largest (or essentia~ largest) element.
show that 3
1
Exa.nwle l(a):
always has llEJdmaJ. elements.
Let 'I
Clearly, the
In section 7 we shall
Consider the two examples •.
consist of the four points
a, b, c and d and let
P consist of only one probability measure--the one that allots equal
probabilities to the four points.
Let lB
consist of the four sets
41 , I , [a, b] and [c,d] •
Then the two sub-fields Cl and C2 ' consisting ,respective~,of
C :
l
C :
2
<1>,
'I, [a,c] and [b,d] ,
<1>,
'I , [a,d] and [b,c] ,
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are each a maximal m-independent sub-field.
Incidentally, in this case C
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and C happen to be independent of each other.
2
EXBJ!!Ple l{b):
Let ~,~, ... x be n independent normaJ. variables with equaJ.
n
unknown means
cp and equal unknwon standard deviations
e.
Let m be the sub-
field induced by the statistic
x=
(~ + ~ + ... + xn)/n
and let C be induced by the set of differences
D = (Xl - Xn , x-~ - xn , ••• xn-I - xn ) •
Here C is m-independent but is not the largeEtCB-independent subfield.
Inde.ed, in this situation, there are infinitely nmly maximal elements
in :;1 (see example 1 in [1]).
It is, however, possible to show that C
is an essentially ma.xi.mal element in 3
reverse the role of
x
• . In the above example, one rmy
1
and D and ask oneself as to whether
statistic that is D-independent.
x
is a maximal
It is of some interest to speculate about
the truth or falsity of the following general
Proposition 1:
If C be a :max:imal (or essentially ma.xi.maJ.) m-independent
SUb-field, then CB is a ma.xirnal (or essentially ma.ximaJ.) C-independent
sub-field.
!±.r..... cp-free
sub-fields (32 ):
Let us suppose that the members of the class p are indexed by two
independent parameters
p
e
and cp, i. e. ,
= ( Pel
,cp e
€
El , cp
€
~)
,
the parameter space being the Cartesian-product El x
4
~
•
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A sub-field C is called cp-free if the restriction of P to C leads to
a class of probability measures that may be indexed by 9 alone, i.e. for all
the probability P 9 (C) is a function of 9 only. Let :;2 be the
.,ep
,
family of all cp-free sub-fields. Evidently, the notion of cp-free SUb-fields
C € C
is a direct generalisation of the notion of ancillary sub-fields.
The trivial sub-field is again the least
eleme~t
alvlaYs has ma.xima.l elements will be demonstrated later.
of:;2.
That 3
In general, 3
2
2
has
a plurality of ma.xi.mal. elements.
Let I
Ex.a.m,ple 2(a):
let P
= (Fe ,cp)
where
e,cp (x)
0 <
a, b, c, d and e and
consist of the probability measur es
x
P
consist of the five points
e<1
a
b
c
1-9
e cp
9cp
e
d
9( t
- cp)
9( ~ - cp)
and 0 < cp < ~ •
There are exactly J2 sub-sets of I whose probability measure is cpfree, and they are I, [a], [b,d], [b,e], [c,d], [c,e] and their complements.
As these J2 sets do not constitute a sub-field it is clear that there cannot
exist a largest element in :;2.
The two sub-fields C and C consist ing
l
2
respectively of
C
l
C
2
I , [a], [b,d], [c,e] and their complements
I,
[a], [b,e], [c,d] and their complements,
are the t-ro maximal elements of :;2.
Ew:sple 2 (b ) :
Let
xl'~' ••• , x
n be n independent and identically distributed
variables with a cumulative distribution function (edt) of the tY1>e
,
-oo<cp<oo,O<9<oo ,
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where the function F is known and cp, a are the so-called location and scale
parameters.
The sub-field C generated by the n-1 dimensional statistic
D
= (~ ~
xn , x-~ - xn , ••• xn-1- xn )
is cp-free in the sense defined before.
is the largest element of the
fami~
In general, it is not true that C
"2 of cp-free sub-fields.
In the particular
case, where F is the cd! of a normal variable, the sub-field C may be shown
to be an essentially maximaJ. element of "2.
Let us observe that, in this
particular case, "2 is the same as "1 of example l(b).
The following proposi-
tion may well be true.
Proposition 2:
Whatever nay be F, the sub-field C (as defined above) is an
essentially max:Lmal element of the family "2 of cp-free (cp being the location
parameter) sub-fields.
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Suppose in example 2 (b) we reverse the role of cp and a and concern
ourselves with the fami.~ ,,; of e-free sub-fields, i.e. with sub-fields every
member of which has a probability measure that does nat involve the scale
parameter a.
The author believes that the following proposition is generally
true.
Pra,position 3:
Every a-free sub-field is also cp-free, i.e.
,,;c:
"2 • (In
the particular case where F is the cd! of a normal variable, the truth of
Proposition 3 has been established in [4].)
5.
q-similar sub-fields
Let q = {g}
("3):
be an arbitrary (but fixed) class of measurable trans-
fornations of (I, a) into itself.
For each Pe P, the transforation g e
1
induces a probability measure Pg- on (I, a).
6
q
A sub-field C will be called
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<i-similar if, for each g
P and PgC
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q
and P e: p , the restriction of the two measures
to C are identical.
In other words, 0 is <i-similar if for all
C
€
for all P e: P
P g-l(C) == p(C)
Let
':Y,
be the family of all Q-similar sets.
g
and
€
<i •
One may look upon
3, as the
family of sub-fields that are induced by statistics T(X) such that T(X) and
T(g x) are identicaJ.ly distributed for each Pe: P
element of
3,
3,
and
g€
q.
The least
is, of course, the trivial sub-field As we shall see later,
always has maximal elements and, in general, a plurality of them.
Example, (a):
Let '1
be the real line and P
=
(Pal - 00< a <00) , where
uniform distribution over the interval (a, a + 1).
Pais the
Let Q ccnsist of the single
transformation g defined as
g x
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€
= the
fractional part of x •
It is easy to check that for all a in (-00 ,00)
PeS
-1
= Po
•
In this example, the sub-field C is q-similar if and only if each member of
C has a probability that is a-free.
Thus, the family
3,
fields is the same as the family of ancillary sub-fields.
of Q-similar subHere,
3, has a
largest element and that is the sub -field of all Borel-sets A such that the
t"l0 sets A and A+l are essentially equal with respect to the Lebesgue measure.
ExaJl!Ple , (b) :
Let ('1 ,
a,
p) be as in example 2 (b) where F is known and
cp, a are the location and scale parameters.
g
a
as
7
Define the shift transformation
I
where
a
is a fixed real number.
= {ga 1-
~
Let
00 < a < 00) be the class of
all shift transformations.
Denoting the joint distribution of (~,~, ••• x n ) by Pq>,
e
,we note
at once that
-1
P
g
cp,e a
= P
q>+a,e
In this example, the family 3
the family 3
2
3
of q-similar sub-fields is the same as
of cp-free sub-fields.
Let us call the set A
g
-1
A
~-invariant
=A
if A€
a
for all
and
g€
~
•
Likewise, let us call A a.1.Joost (}-invariant i f the two sets g
A are . . p-equivalent for all geq. Let lB
class of (}-invariant and almost
<B. and (B
J.
~-invariant
are members of the family 3
a
and lB
i
3
sets.
a
-1
A and
be, respectively, the
It is easy to check that
of (i-similar sub-fields.
The follow-
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ing proposition should be provable under sone conditions.
Proposition 4:
~-similar
The SUb-field lB
a
of al.m::>st (}-invariant sets is a maximal
sub-field.
Under sone general conditions it should also be true that the subfield lB
i
of (}-invariant sets is an essentially maximal element of 33"
This
is so in the case of example 3(b) where F is the cdt of a normal variable.
6. (B-linked sub-fields (34 ):
Let <B
linked if
a
be a fixed sub-field of G.
A sub-field C will be called lB-
be sufficient for (C,p), i.e., for every C
€
C , there exists
a <B-measurable mapping Q(C,·) of 'I into the unit interval such that, for al.l
P(BC)
=
J:
Q(C,·)d p(.)
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Let
~4
The trivial sub-field is
be the family of allm-linked sub-fields.
again the least element of
~4.
We shall presently see tmt
~4
always has
maximal elements.
Exa.n!Ples 4(a):
(i)
Let m be the trivial sub-field.
that, in this instance,
~4
It is easy to see
is the same as the family of all ancillary sub-
fields.
Let us suppose that P is indexed by the parameters cp
(ii)
and
e.
Let m be a fixed cp-free sUb-field, i.e., a lnember of
in section 4.
In this instance, every
(iii)
~2
as defined
CB-linked sub-fields is also cp-free.
Let m be a sufficient sub-field.
In this case
~4 is
the family of all sub-fields.
Example 4(b}:
Let (X , G., p) be as in example l(b) and let CB
field induced by the sample variance
E(X
i
-
X)2/ n
•
o
be the sub-
If C be the sub-field
induced by
D
= (~-x
,
.L n
~-x , ••• , x 1- x ),
i::: n
nn
then it is easy to check that C is CBo -linked.
Since m0 is rn-free
it follows
T
that every m -linked sub-field is also cp-free.
o
It is possible to show that
C is an essentially ma.x:imaJ. m -linked sub-field.
o
The truth of the follow.i.n.g
proposition is worth investigating.
Proposition 5:
If CB
o
and C be as in example 4(b), then C is an essentially
largest element of the family
7.
~4
of the CB -linked sub-fields.
o
Existence of maximl elements:
In this section we develop sone general methods to prove the existence
of nwdmal elements in the families 3 , 3 , 3 and 3 • Let us first nate a
2
4
1
3
common feature of the four families of sub-fields. Each~. (i
1,2,3,4) is
J.
9
=
I
the totality of all sub-fields that can be embedded in a certain class
of measurable sets.
ei
This will be clear once we define the four classes
e , e , e and e4 of measurable sets.
2
l
3
Definitions:
(i)
i.e.
e1 -(ii)
i.e.
i. e.
be the class of all (B-independent (see section 3) sets,
= P(A)P(B)
(AI p(AB)
e2
Let
, for all PEP, B €<B}
be the class of all cp-free (see sect:ion 4) sets,
e2 = (A Ip<p, e(A) does not involve cp}
(iii) Let e
be the class of all <i-similar (see
3
e3 = (A I p(g-lA) = P(A) for all P € P , g € Q} •
(iv)
i. e.
el
Let
A €
section 5) sets,
Let e4 be the class of all <B-linked (see section 6) sets,
e4
if and only if there exists a <B-measurable mapping Q(A,.) of
1. into the unit interval such that
~
p(AB) =
Q(A,· )dP(·) for all P€ P and B € m •
= 1,2,3,4.
It is now clear that, for i
3. = (C I C is a subfield and C c e.)
1
i. e. ,
ei
,
1
:Ji is the family of all sub-fields that can be embedded in the class
of measurable sets.
Our first general result is the following:
Theorem 1:
Each
(a)
,
(b)
A
(e)
ei
ei
€
€
ei
(i = 1,2,3,4) has the following properties
ei
,
' X
B
€
€
ei
ei
' Ac B
===> B -
A
€
ei
•
is closed for countable disjoint unions.
The proof of Theorem 1 is routine and hence omitted.
An immediate
consequence of Theorem 1 is the
Corollary:
Each
ei
(i
= 1,2,3,4)
is a mnotone class of sets.
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The follov:i:ng is our fundamental existence theorems.
Theorem 2:
If
e
be a given monotone class of sets and:r be the family of a.ll
Borel-fields that could be embedded in
e. ,
then corresponding to each element
.""
,...
C of :; there exists a ma.x:i.ml element C of :; such that C c C •
Let {Ct It€ T} be an arbitrary linearly ordered (wrt the partial
Proof:
order of inclusion relationship) sub-family of :J and let
is linearly ordered it follows that C
Since
sets.
o
The monotone extension of C
o
Cl of Co.
Since
hence C € :;.
l
bound in:J.
e
is a field of
is then the same as the Borel-ext ension
is monotone and Co c
e,
it follows that C c:
l
e
and
Thus, every linearly ordered sub-family of :; has an upper
Theorem 2 is then a consequence of ZOrn t s Lemma.
An immediate consequence of Theorem 1 and 2 is
Theorem 3:
that C c:
8.
For each C €:;i there exist s a maximal element
C,
C
in:Ji such
(i = 1,2,3,4).
Some general results:
Let
e.
be a class of measurable sets having the same characteristics
as those of the classes
a)
~ €
b) A €
c)
e
e. i
e.
e , 'I €
e, B €e,
in Theorem 1.
A
That is,
c B ==> B - A
€
e.
•
is closed for countable disjoint unions.
Let :; be the family of all the sub-fields that may be embedded in
e
and let:;
o
be the sub-family of all the maximal elements in :;.
is not vacuous, has been esta.blished in Theorem 2.
11
That:;
0
I
Two members A and B of
A
of
e
€
is said to be
are said to 'conform' 1f AB
conforming t if AB
€
e
for all B
€
e.
€
e.
The set
If every member
e be conforming, then e must itself be a Borel-field; and hence there is
no problem as 3
is
t
e
t
consists of a single member Il8JIEly
0
e
itself.
A sub-field
conforming t if every one of its members is so.
e,
e•
Let , be the class of all the conforming sets in
~
A €
Let
.ft\,
element of 3.
==> A
€
e
and AB€
e
for all B €
stand for a typical element of 3
0
,
i. e.
~
i. e.
is a maximal
We prove the following
Theorem 4
i) .M. is a maxilDal. element of 3 I if and only if, A
that A does not conform to at least one member of
~
€
e
-./l1. implies
•
8 is a sub-field and is equal to the intersection of aJ.l the
ii)
ma:ximaJ. elements in 3.
It is the largest conforming sub-field.
C is a conforming sub-field) if and only if, for all <B
iii)
true that C V <B
€
3.
€
3 it is
[C V <B stands for the least sub-field oontaining both
C and <B.]
Let X
~:
€:J
o
and let A be a fixed member of
let A conform to all the members of
~.
e-.M • If possible,
Consider the class
*
))t
of sets
of the type
AM:t U At~
where
l\
,1"" *
€:J
and
~
,
are arbitrary members of oM..
and that A € M.
*
and.M. c
that.M. is a maximal element of :J.
)1.
*•
It is easy to check that
This violates the supposition
Thus, the 'only if' part of (i) is proved.
To prove the 'if' part we have only to observe that i f J1 is not maximal then
* e
there exists a larger sub...f"ield.,4t. c
an A
€
e
and this implies the existence of
-.M. that confonns to every member of .}t.
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Since, every member of
e,
fj
conforms (by definition) to every nember of
it is an iImnediate consequence of (i) that
fj
c: Jtt for each
Itt.
€
:J 0
i. e.
'
•
Now,let
M and E be typical members of
nM
From Theorem 2, there exists a ma.x1ma.l element oM,
sub-field consiting of
sub-field Itt
that
o
M € fj.
~,
E, E' and 'I.
and hence they must conform.
and
in:J which CQ'ltains the
o
Thus, M and E are together in the
Since E is arbitrary, it follows
We have thus proved the equality of 8 and
incidentaly proved the equality of
it is now clear that
fj
= nX
e
e respectively.
and U JIl •
n.M.
and has
Since each oM is a sub-field
is also a sub-field.
That it is the largest
conforming sub-field, follows from its definition.
Now, let C be an arbitrary conforming sub-field i.e. let
sub-field of 8.
For each
(B€
.M. of CJ such that CB c: tN..
i.e.
C V CB
€
:J.
C be a
3 , there exists (Theorem 2) a anaximal element
But C c:
fj
c: Jot. • Therefore, C
This proves the 'only if' part of (iii).
V
CB c: J( c:
e,
The 'if' part
is trivial.
For example, let
e
be the class of all CB-linked sets (see Sections 6
and 7) in the probibility structure ('I, a, p) where CB is a fixed sub-field
CB-linked, i.e., if there exists a
of a. If the set A be!m-measurable function Q(A,·) satisfying definition
(iv) of Section 7, then it is easily seen that AB is CB-l1nked for every
B
€ (B.
We have only to define Q(AB,·) as
Q(A,·) I (B,.)
where I(B,·) is the indicator of B.
In this case, CB is a conforming sub-field.
Theorem 4(iii) then asserts
that for every CB-linked sub-field C the sub-field CB V C is also CB-linked.
of
It will bel some interest to find out conditions under which <B is the largest
I
conforming sub-field, i. e. CB
••I
=~ •
Some further problems
,2.
In this section
(A)
'We
list four problems that are mostly unsolved.
Separating sub-fields:
Let P be a class of 'distinct t probability
measures on a measurable space
(x.,
a).
That is, for each pair Pl'P of
2
members of p there exists a measurable set A € a such that P (A) f:. P2(A) •
l
A sub-field CB will be called t separating t if the restriction of p to CB gives
rise to a class of distinct measures.
field is separating.
separating.
tion
a
Let 3
5
For example, every sufficient sub-
No ancillary or cp-free (see Section 4) sub-field is
be the family of all separating sub-fields.
is the largest elenent of 3
5
By defini-
• What can we say about the existe nce
of minimal elements in 3 ? A variant of this problem has recently received
5
some attention in USS:! ([6] and [8]). A partition IT of 1. into a class of
disjoint measurable
sets (At)
will be called 'separating t if for each
pair Pl'P of member of P there exists a member At of the partition IT such
2
that P1 (At)
f:.
P2 (At).
A separating partition is called minimal if there
exists no other separating partition with a smaller number of parts.
v
Let
(p) stand for the number (possibly infinite) of parts in a minimal separat-
ing partition.
Eean'Wle 5(a):
What can we say about v(P)?
Consider the class P of all normal distributions on the real
line with unit variances.
Here
v(p) = 2.
Any partition of the real line
into two half lines is clearly separating and, of course, minimal.
corresponding sub-field is a minimal element of 3
Exam,ple 5(b):
° < 9 < 1.
5
The
•
Let P be the fa.m1ly of uniform distributions on [0,9],
In this case v (p)
=3
(see [6]) •
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Example 5( c ) :
then
vCr) < n
then
vCr)
(B):
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•
= 2 •
If p consists of a countable number of continuous measures
(see [6] and [8]).
Partially sufficient SUb-fields:
The notion of partial sufficiency as introduced by Fraser
}
([5])I
is
as follows.
Let p
=
(p
cp,
e)'
cp
€ <ll ,
e
€
8 , be a family of probability measures
indexed by the two independent paramters cp and
u (or
be called e-sufficient for
e.
A sub-field <B c
a
will
simply e-sufficient) if
i) <B is cp-free in the sense of section 4 and
ii)
for each A€
u,
t11ere exists a choice of the coo.ditional probability
(function) of A given <B tl18,t does not depend on
there exists a <B-measurable function
e,
i. e. ~ for each
m
T
€ <ll
1
\ (A, .) that maps X. to the unit intervaJ.
in such a manner that
for all B
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If P consists of a finite number of measures P ,P , ••• P
1 2
n
Let
€
<B
~6
and
e€e.
be the family of all e-sufficient sub-fields.
cond.itions can we prove that 3'6 is not vacuous?
maximal elements in
(C):
~6
Under what
What about the mini.ma.l and
?
C05Plete sub-fields:
Given a probability structure
(x., u, p), we call a sub-field <B 'complete'
if, for a <B-measurable, p-integrable function f, the integral
r
f d P
f
is p-equivalent to zero.
jx.
when and only when
complete sub-fields.
== 0, for all P
€
P ,
Let 'J.
7
be the family of all
What can we say about the existence of maximal and
minimal elements in 3 ?
7
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Let us terminate this list of problems with a finaJ. one.
.(p) :
Conw1ementary sub-field
Let (X , G) be a gi.ven measurable space and let CB be a fixed sub-
field of G.
A sub-field C will be called a complement to CB if
CBVC=G,
i. e., if G is the least Borel-field that contains both m and C •
Let 3
8 be the fa.mi.l.y of aJ.1 sub-fields that are complements to CB •
For example, if CB is the triviaJ. sub-field thfLn 3
element namely G itself.
If
8
consists of a single
m = G , then 3 8 consists of
all sub-fields of
G •
G is, of course, the largest element of 3 8 •
examples where 3
8
It is easy to construct
has a multiplicity of minimal elements.
have a minimaJ. element?
Does 3
8
always
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References
1.
Basu, D. (1959):
247-256.
The family of ancillary statistics, Sankhya, ~
2.
Basu, D. (1965): On maximal and minimal sub-fields of certain types,
Institute of Statistics Mimeo Series No. 422.
3.
Burkholder, D. L. (1961): Sufficiency in the undominated case, Ann.
~. Stat., ,2g, 1191-1200.
-
4.
Dantzig, G. B. (l9!n): On the non-existence of tests of Students I
hypothesis having power functions independent of 0" , Ann. ~. ~.,
g, 186-191.
5.
Fraser, D. A. S. (1956):
6.
Kagan, A. Me and Sudakov, V. N. (1964): Separating partitions of certain
families of measures (in Russian). Vestnik Leningrad University, No. 13,
&me
~. ~.,
Sufficient statistics with nuisance parameters
gz, 838-842.
147-150.
7.
Pitcher, T. S. (1957): Sets of measures not admitting necessary and
sufficiency statistics or sub-fields, Ann. ~. ~., ~ 267-268.
8.
Visik, S. Me, Cobrinksii, A. A. and Rosenthal, A. L. (1964): A
separating partition for a finite family of measures (in 1Ihlssian),
Theory of Probe and its application, ,2, 165-167.
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