Hasegawa, Minoru

Perlman, Michael D.

University of Minnesota Digital Conservancy

~ 
.. 
.., 
'-* 
... 
-
~ 
.... 
~ 
-
-
-
'-
lad 
ad 
._ 
.. 
... 
~ 
ON THE EXISTENCE OF A MrNIMAL SUFFICIENT SUBFIELD 
By 
Minoru Hasegawa and Michael D. Perlman1 
University of Minnesota, Minneapolis 
Technical Report No. 210 
May, 1973 
Abbreviated Title: Minimal Sufficient Subfield 
1 Research supported in part by National Science Foundation Grant No. GP-34482 • 
-ABSTRACT 
On the Existence of a Minimal Sufficient Subfield 
Let (x, s) be.a measur~ble space and M a collection of probability 
measures on s. Pitcher (Pacific J. Math, (1965), p. 597-611) introduced a 
condition called compactness on the statistical structure (X, S, M), more 
general than domination by a fixed cr-finite measure. Under this condition 
he gave a construction of a minimal sufficient subfield. In this paper a 
counterexample invalidating this construction is presented. We give a non-
constructive proof of the existence of a minimal sufficient subfield under 
a condition slightly weaker than compactness. The proof proceeds by considering 
the intersection of an uncountable collection of sufficient subfields, and 
relies on a martingale convergence theorem with directed index set, due to 
Krickeberg. 
American Mathematical Society 1970 subject classification. 
Primary: G2B05. 
Key words and phrases: minimal sufficient subfield, dominated family, 
discrete family, compactness, coherence, interse~tion of sufficient subfields, 
martingale. 
- 1 -
-1. Introduction 
Let S be a a-field of subsets of a space X and let M be a collection 
of probability measures on (X, s). A sub-a-field (abbreviated hereafter as 
"subfield") Tc S is said to be sufficient for M if for each A in S, 
there exists a T-measurable function f(x) such that f = P[AIT] for each 
P in M (f depends on A but not on P and may be chosen so that 
0 ~ f ~ 1). All subfields considered in this paper are subfields of S. 
A set A in S is called M-null if P(A) = 0 for each P in M. 
Let N denote the subfield consisting of all M-null sets and their complements. 
For two subfields Tl, T2 of s, write T1 ~ T2 [M] if for each Al in Tl 
there exists A2 in T2 such that (A1 - A2) U (A2 - A1) is M-null. Note 
that T1 ~ T2 [M) if and only if T1 :: T2 VN, where T2 VN denote the 
subfield containing T2 and N. Following Bahadur (1954) we say that a 
subfield T0 is minimal sufficient for M if T0 is sufficient and if 
T0 :: T [M] for any other sufficient subfield T. 
smallest 
If there exists a a-finit~ measure A such that every P in M is 
absolutely continuous with respect to A we speak of the dominated case, 
which has been studied by Halmos and Savage (1949) and Bahadur (1954). 
If every P in M is a discrete measure, if S is the collection of all 
subsets of X, and if N = (~, X) we speak of the discrete case, which 
has been studied by Basu and Ghosh (1969) and Morimoto (1972). In both the 
dominated and discrete cases, sufficiency can be characterized by a Factorization 
Criterion, and by means of this criterion a minimal sufficient subfield can 
be constructed (see Bahadur (1954) and Basu and Ghosh (1969)). 
- 2 -
--
._ 
• 
-
-
tel. 
~ 
-
1W 
qp 
-
.. 
-
-
-
-
.. 
... 
Pitcher (1965) introduced a more general condition on {X, s,- M), which 
he called compactness. The compact case includes both the dominated case 
{Pitcher (1965), Theorem 1.2) and the discrete case {Kusama. and Yamada (1972), 
Theorem 1.1) and is not exhausted by these two cases (Pitcher (1965), Example 1)., 
In his Theorem 2.5 Pitcher (1965) states that under the assumption of compact-
ness on {X, S, M) there exists a minimal sufficient subfield and in the 
proof claims to construct such a subfield. This proof is in error, however, 
as we show in section 5 of this paper. 
The main purpose of this paper is to provide a veracious proof of a 
slight generalization of Pitcher's Theorem 2.5, demonstrating the existence of 
a minimal sufficient subfield under a restriction on {X, S, M) slightly 
weaker than compactness. Our proof differs from that attempted by Pitcher in 
that it is non-constructive and relies on a convergence theorem of Krickeberg 
(1960) for a martingale with a directed index set. Based on our discussion 
in section 5, we feel it is unlikely that a constructive proof can be found. 
In section 2 we outline our approach and state the main result, 
Theorem 2.3. Section 3 contains some preliminary results needed for the 
proof of the main theorem, which proof is given in section 4. In section 5 
we present an example to show that Pitcher's construction of a minimal sufficient 
subfield is incorrect. 
- 3 -
--
2. The existence of a minimal sufficient subfield. 
In the dominated and discrete cases a minimal sufficient subfield can 
be constructed using the Factorization Criterion for sufficiency. In moTe 
general cases where one does not have such a criterion available, it is 
natural to investigate the existence of a minimal sufficient subfield by 
asking whether the intersection of a collection of sufficient subfields is 
sufficient. Our starting point is a result of Burkholder (1961, Theorem 4 
and Corollary 2) which states that without any restrictions on (X, S, M), 
the intersection of a finite or countable collection of sufficient subfields 
is sufficient, provided that each contains N (Burkholder also gives an 
example to show that the result is not true without the proviso concerning N). 
Theorem 2.1.(Burkholder). Let (T. I i = 1, 2, ••• ) be a finite or countable 
l. 
collection of sufficient subfields, for M such that 
Then n T. is also sufficient for M. 
N c T. 
]. 
for each T .• 
]. 
]. 
Furthermore, in the dominated case Theorem 2.1 is true for an arbitrary 
(not necessarily countable) collection: 
Theorem 2.2. Suppose M is dominated. Let {Ta) be an arbitrary collection 
of sufficient subfields for M such that 
is also sufficient for M. 
NCT 
a 
for each T. 
a 
Then 
* * Proof. Let T be a minimal sufficient subfield for M. Then T :: Ta[M] for 
* * * each a, i.e., T c T VN = T"'. 
- a u; 
Hence T :: n Ta. Since T is sufficient 
and M is dominated, this implies that n Ta is sufficient (Bahadur (1954), 
Theorem 6.4). 
A similar argument shows that Theorem 2.2 is true in the discrete case. 
Dropping the assumptions of dominance or discreteness, suppose that, conversely, 
- 4 -
--
-
the conclusion of Theorem 2.2 is satisfied for (X, S, M). Then a minimal 
* sufficient subfield T exists. To see this, simply let 
collection of all sufficient subfields which contain N 
{T} be the (X 
(s is one such 
subfield), * * and let T = n T. By our supposition, T is sufficient. 
a 
If T is any other sufficient subfield then TVN is also sufficient and 
contains N, hence belongs to * * Thus T = n T c T V N, i • e. T c_ T [M] , 
(X -
·>f-
so T is minimal sufficient. 
The conclusion of Theorem 2.2 does not hold without some restriction on 
(X, S, M), for Pitcher (1957) has shown that a minimal sufficient subfield 
does not always exist. Our main result, Theorem 2.3, states that if (X, S, M) 
is coherent (Definition 2.3) then the conclusion of Theorem 2.2 holds and a 
minimal sufficient subfield exists. This result includes Theorem 2.5 of 
Pitcher (1965). 
Let B be the collection of all S-measurable functions f such that 
0 ~ f(x) ~ 1 for all X in x. For each p in M let B(P) be a copy 
of B, and let TTB(P) denote the Cartesian product of the B(P) as p 
ranges over M. An element of 1TB(P) is denoted by (fp). 
Definition 2. 1. An element (fp) of TTB(P) is said to be finitely coherent 
(countably coherent) if for each finite (countable) subset cp = {P.} of M, 
1 
there exists a function f in B such that f = fp_ a.e. [Pi] for cp cp 
1 
each P. in cp. 
]. 
Definition 2.2. An element (fp) of TTB(P) is said to be coherent if there 
exists a function f in B such that f = fp a.e. [P] for each p in 
Remark. The notions of finite coherence and countable coherence coincide. 
- 5 -
M. 
.... 
For , if is finitely coherent and if cp = (P.) is a countable subset 
l. 
of M, we can define fcp by 
{
, lim f 
f (x) = cpn 
cp 
0 
if the limit exists 
otherwise 
where cpn = (P1 , ... , Pn). A countably coherent element (fp) need not be 
coherent - see the Example at the end of this section. 
Definition 2.3. (x, S, M) is said to be coherent if every countably coherent 
element (fp) of ITB(P) is coherent. 
The following is our main result. Its proof is given in section 4. 
Theorem 2.3. Suppose that {X, S, M) is coherent. 
Let {Ta) be an arbitrary collection of sufficient subfields for M such that 
Then n T is sufficient for M. 
Ci 
Thus, if (X, S, M) 
is coherent there exists a minimal sufficient subfield. 
If (X, S, M) is compact according to the definition of Pitcher (1965, p.598) 
then it is coherent. This follows easily from the statement at the top of 
p. 599 and Lemma 1.2, both in Pitcher (1965). It is easy to verify directly 
that {X, S, M) is coherent in the discrete and dominated cases. 
is dominated, for example, then M contains a countable subset 
If (X, S, M) 
* cp = {P.) 
1. 
which is equivalent to M (Halmos and Savage (1949), Lemma 7). Suppose that 
(fp) is countably coherent. We claim that fcp* = fp a.e. [P] for each P 
in M, so is coherent. To see this, let 
f = f = f * a.e. [Pl..] 
cp pi cp * for each Pi in cp, 
Hence a.e. [P], as required. 
- 6 -
* q, = {P) U cp • Then 
so f = f * a. e. [M]. 
cp cp 
--
Finally, consider the following example of Pitcher (1965, Example 2). 
Let X be [O, 1), S the Borel a-field on X, and M the collection 
of all measures on S which are either degenerate at a single point of X 
or else are absolutely continuous with respect to Lebesgue measure. For 
each P in M and x in X define fp(x) = P({x)). Then (fp) is a 
countably coherent element of rrB(P) but is not coherent. Thus (X, S, M) 
is not coherent. However, S itself is minimal sufficient: for any sufficient 
subfield T and any A in S, let g = P[AIT] for all P in M. If 
P is the probability measure degenerate at x, then 
X 
for all x, where IA denotes the indicator function. Since g is T-measurable, 
A must belong to T, hence T = S, so S is minimal sufficient. This 
example shows that coherence is a sufficient but not necessary condition for 
the existence of a minimal sufficient subfield. Nonetheless, we think it 
unlikely that a significantly more general condition for the existence of 
a minimal sufficient subfield can be found. 
- 7 -
-3. Preliminary results. 
4 
In the course of the proof of Ti:ieorem 2. 3 we shall need the fact 
(Theorem 3.2) that in the dominated case, Theorem 2.2 remains true if the 
proviso that N c T for each T is replaced by the assumption that 
a a 
{Ta) is directed downward by inclusion (defined below). Without the assumption 
that N c T, the proof of this assertion is substantially more difficult 
a 
than the simple proof of Theorem 2.2. (In Theorem 3.2 the collection of sufficient 
subfields is denoted by {Ut) rather than {Ta).) 
We shall rely on a reverse martingale convergence theorem of Krickeberg 
{196o), for which the following terminology is needed. Let (X, S, µ) be 
a probability space. Let ® be an arbitrary index set and let (UT I Tin®) 
be a non-empty collection of subfields of S directed downward by inclusion, 
Le., for each pair a, T in ® there exists p in ® such that 
U c U n U. Then ® becomes a directed set under the partial ordering p - a T 
< < defined by p < < T ~ U c U • For each T in ® let hT be a real-
T - p 
valued, U -measurable, 
T 
µ-integrable function. 
is said to be a martingale (relative to µ} if 
h = E [h I u] 
T µ p T a.e. [µ] 
The collection { (h , U ) I T in ®) T T 
whenever p < < T. The collection 
if 
(h) 
T 
is said to be µ-uniformly integrable 
J lhTI dµ-+ 0 
(lh'Tf>v) 
- 8 -
as y ~ oo 
.... 
uniformly in rr. Let II h IIµ, denote the L1(µ,)-norm of h, and set 
U-oo = n UT. As a special case of Theorem 2.2 of Krickeberg (1960), we 
state the following result. 
Theorem 3.1. (Krickeberg) Let {(h, U) f Tin®} be a µ-uniformly 
'r T 
integrable martingale. Then there exists a 
function h such that 
-00 
lim II h - h IIµ, = o. 'r -co T 
Theorem 3.2. Suppose M is dominated. Let 
U -measurable, 
-co 
µ-integrable 
{U} be an arbitrary collection 
T 
of sufficient subfields for M such that {U} is directed downward by inclusion. 
T 
Then U = n U is also sufficient for M. 
-co 'r 
Proof. Since M is dominated, there exists a countable subset (Pi} c M 
such that every P in M is absolutely continuous with respect to 
"' -i A =t12 P •• A well-known factorization criterion for sufficiency in the 
1. 
dominated case states that a subfield T is sufficient for M if and only 
if for each P in M there exists a T-measurable version of the Radon-
Nikodym derivative dP/dA on S (Halmos and Savage (1949), Bahadur (1954)). 
Fix P in M. Since each U is sufficient, there exist nonnegative 
T 
UT-measurable versions fT of dP/dA on S. For any p, rr we have 
(4.1) 
for all A in S, so {(fT, UT)} is a martingale relative to A. 
Furthermore, (4.1) implies that 
(4.2) f = f 
T p a.e. [A]. 
- 9 -
Since J fTdA = 1 < oo we have that 
as y ~ oo 
and by (4.2) this convergence is uniform in T, 
integrable. 
such that 
Thus by Theorem 3.1 there exists a 
lim 
T 
- f 
-00 
In particular, for each A in S we have 
JA f-oo dA = lim JA fT dA = P(A), 
T 
so {f} is A-uniformly 
T 
U -measurable function 
-00 
f 
-00 
so f is a U -measurable version of dP/dA on S. Since this holds for 
-00 -00 
each P in M, U = n U is sufficient, which completes the proof. 
-co T 
We shall also need the following lennna, which is similar to Theorem 2.3 
of Pitcher (1965). For each P in M let ~ denote the subfield of all 
P-null sets and their complements. For each subfield T, define 
T = n (T vN ) ; 
Pin M p 
T is also a subfield of S. It is easy to see that a function f"" is 
T-measurable iff for each P in M there exists a T-measurable function 
fp such that f = fp a.e. [P]. Clearly, Tc TV N c T for each subfield T. 
Lemma 3.3. If T is a sufficient subfield of S, then TVN = T. 
- 10 -
... 
Proof. Suppose f. is T-measurable and bounded. Since T is sufficient 
there exists a bounded T-measurable function f such that f = Ep [t I T] 
for each P in M. Since 7 - f is T-measurable, for each P there 
exists a T-measurable function fp such that t - f = fp a.e. [PJ. Therefore, 
for each P, 
J (f - f) 2 dP = J ( f - f) fp dP 
= o. 
Hence .£ = f a.e. [P] for each P. Since f is T-measurable, this implies 
that f is TVN-measurable, completing the proof. 
,.. 
Remark. The subfields T in our paper replace the T considered by Pitcher 
,.. 
(1965, p.601). T and T are defined similarly, except that TC S whereas 
,.. 
Pitcher, by considering outer measures, does not restrict T to be contained 
in s. There does not appear to be any essential difference in the two treatments. 
- 11 -
--
-
4. Proof of Theorem 2.3. 
Let {UT} be the collection of all finite intersections formed from 
the given collection (Ta}. By Theorem 2.1 each subfield UT is sufficient 
for M and contains N. Furthermore, 
hence is directed downward by inclusion. 
(U) is closed under finite intersections, 
T 
We are to show that U - n U = n T 
-oo T a 
is sufficient for M. For this, we fix A in S and will find a U -measurable 
..00 
function f such that f = P[A U . ] for every P in M. 
..00 
For each p in M let 0 s; fp s; 1 be a version of P[A I u 1. We 
..00 
claim that {fp) is countably coherent. To see this, let cp = {P.} be a 
1 
countable subset of M. Each u 
'1" 
is sufficient for cp, and cp is a 
dominated family, so by Theorem 3,2, u is sufficient for cp • Hence there 
...00 
exists a U -measurable function f such that f = P. [A 
-00 cp cp 1 U ] for each -00 
Pl.. in cp. Thus f = f a.e.[Pl..]; hence (fp) is finitely coherent. 
cp pi 
Thus, by the hypothesis that (X, S, M) is coherent, there exists an S-measurable 
function Os: f s: 1 such that f = fp a.e.[P] for each P in M. 
Since each fp is U -measurable for every T, 
'r 
the preceding implies 
that f is U -measurable. However, by Lemma 3. 3, U = U VN = U • Thus 
T T T T 
f is U -measurable for every T, hence f is U -measurable. Since 
T ..oo 
f = fp a.e. [P] for every P in M, this implies that f = P[A I U-oo] 
for every P, as required. 
- 12 -
.... 
-
-
5. A counterexample to Pitcher's construction of a minimal sufficient subfield. 
Pitcher (1965) denotes probability measures in M by µ,, v, etc. 
In the proof of his Theorem 2.5 he begins the construction of a minimal sufficient 
subfield as follows: "Let T0 be the subfield generated by all the functions 
[ dµ, / d{µ, + v)] for µ, and v in M''. Then, using his assumption that 
(X, S, M) is compact (coherence would suffice) he correctly shows that the 
,. 
subfield T0 (or, T0 in our treatment) is sufficient. Lastly, he claims that 
,. 
T0 is minimal sufficient - however, this need not be true. 
The difficulty is that the Radon-Nikodym derivatives dµ, / d(µ, + v) are 
well-defined only up to sets of (µ, + v)-measure O, but Pitcher does not 
specify which versions are to be chosen. The following example shows that if 
,. 
the choice of versions is unfortunate, T0 (or 'r0 ) need not be minimal sufficient. 
Let X be the interval [O, 3], S the Borel a-field on X, and 
where µ,. 
1. 
is the uniform probability measure on the interval 
[i-1, i]. One possible choice of version of dµ,1 / d{µ,i + µj) is the indicator 
function With this choice T0 is simply the field generated by 
,. 
the three intervals ro, 1], [1, 2], [2, 3], and To (or To) is easi~y 
seen to be minimal sufficient. Suppose, however, that we choose the versions 
of dµ,. I d(µ,. + µ,.) 
1. 1. J 
to assume the value 1 on [i-1, i], 0 on [j-1, j]' 
and to be linear with positive slope on [k-1, k], where k 
=I= i, j. Then TO 
,. 
is the entire Borel a-field s and TO (or rf0 ) is sufficient, but not 
minimal sufficient. 
In this example the proper choice of version for dµ, / d(µ, + v) is clear. 
However, in the generality assumed by Pitcher and in the present paper, with 
- 13 -
--
-
--
lal 
la! 
lat 
_, 
c.l 
'-' 
4-1 
-
lal 
'-
-
laal 
lad 
la 
al 
"' l•." :;~ !' l • ' 
no topological structue on (X, s), we do not see that a proper version 
can be specified. It was for this reason that we have sought to prove the 
existence of a minimal sufficient subfield by an indirect approach, examining 
the sufficiency of the intersection of an uncountable collection of sufficient 
subfields. 
- 14 -
~-
... 
-· 
REFERENCES 
[1] Bahadur, R.R. (1954). Sufficiency and statistical decision functions. 
Ann.~· Statist. ~ 423-462. 
[2] Basu, D. and Ghosh, J. K. (1969). Sufficient statistics in sampling 
from a finite universe. Proc. 36th Session Interriat. Statist. 
- --------
Inst. (in ISI Bulletin) 850-859. 
-
[3] Burkholder, D. L. (1961). Sufficiency in the undominated case. 
~· ~. Statist. Jg li91-1200. 
[4] Halmos··, P. R. and Savage, L. J. (1949), Application of ~he Radon-Nikodym 
Theorem to the theory of sufficient statistics. ~- Math. Statist. 
20 225-241. 
[5] Krickeberg, K. (1960). Absteigende Semimartingale mit filtrierendem 
Parameterbereich. Abh. Math. Sem. Univ. Hamburg 24 109-125. 
-------= 
[6] Kusama, T. and Yamada, S. (1972). On compactness of the statistical 
structure and sufficiency. Osaka J. Math. ~ 11-18. 
[7] Morimoto, H. (1972). Statistical structure of the problem of sampling 
from finite populations. ~- ~- Statist. ~ 49()-497. 
[8] Pitcher, T. s. (1957). Sets of measures not admitting necessary and 
sufficient statistics or subfields. Ann. Math. Statist. 28 267-268. 
----- == 
[9] Pitcher, T. s. (1965). A more general property than domination for 
sets of probability measures. Pacific J. Math. 12 597-611. 
- 15 -
I 


On the Existence of Minimal Sufficient Subfield

https://conservancy.umn.edu/bitstream/handle/11299/199203/Technical%20Report%20210%20On%20the%20Existence%20of%20Minimal%20Sufficient%20Subfield%20Hasegawa%20_%20Perlman.pdf?sequence=1&isAllowed=y

On the Existence of Minimal Sufficient Subfield

Abstract

Similar works

Full text

Available Versions

University of Minnesota Digital Conservancy