On the measurability and consistency of minimum contrast estimates

On the Measurability and Consistency of Minimum Contrast Estimates

By J. PFANZAGL, K61n 1)

Summary: The concept of minimum contrast (m.c.) estimates used in this paper covers maximum likelihood (m.1.) estimates as a special case. Section 1 contains sufficient conditions for the existence of measurable m.c. estimates and for their consistency.

The application of these results to m. 1. estimates (section 2) yields the existence of m.1. estimates for families of p-measures (probability measures) which are compact metric or locally compact with countable base, admitting upper semicontinuous densities, whereas the classical results refer to continuous densities. This generalization is insofar of interest as upper semicontinuous versions of the densities exist whenever the densities are /a-upper semicontinuous (whereas /~-continuity does not, in general, entail the existence of continuous versions).

Under appropriate regularity conditions, consistency of asymptotic maximum likelihood estimates is proven for compact (and also locally compact) separable metric families of p-measures with upper semicontinuous densities and for arbitrary families having uniformly continuous densities with respect to the uniformity of vague convergence. The conditions sufficient for consistency are shown "indis- pensable" by counterexamples.

Section 3 contains auxiliary results. Besides their relevance for sections 1 and 2, some of them may also be of interest in themselves, e.g. Theorem (3.4) on the selection of semicontinuous functions from semicontinuous equivalence classes.

1. Minimum Contrast Estimates

Let d be a a-algebra over a basic set X and ~ I d a family of p-measures. Let (T, 41) be a topological space and r: ~ 3 ~ T a map of ~ into T. z will be called a parametrization of ~3 and z(P) the parameter value pertaining to P. We do not require the map z to be onto T, i.e. r ( ~ ) : = {r(P): Pe~]3} will, in general, be a true subset of T. We remark that this general concept of a parameter also covers the case that ~3 itself is endowed with a topology (for T = ~, z = identity map).

1) Prof. Dr. J. PFANZAGL, Mathem. Institut, 5 K61n-Lindenthal, Weyertal 86.

250 J. PFANZAGL

Modifying an idea of HUBER [1966] we introduce the following concept:

(1.1) Definition: A family of d-measurable functions ft: X ~ [ - ~ , + ~] , te T, is a family of c o n t r a s t functions Jbr ~3 if P [J~] exists 2) Jbr all P ~ ~, t e T, and if

(1.2) P[ f~w)]<P[ f ] for all t@z(P) and all Pc~3. n

Later on we shall consider expressions ~ J; (xi). As f may assume the values

as well as - ~ , a special convention, namely - ~ + oo:= - ~ , is necessary.

(1.3) Example: Let ~ ] d ~ # l N with/~[~' a-finite and let h e be a density of P [ d with respect to /~]~.

Then, for T = ~ and z = identity map, the family fe: = - log he, P e ~ , is a family of contrast functions, if - ~ < P [log hp] for all P E ~3 and P [log hQ] < oo for all P, Q e ~3 with P 4= Q: If P [log hp] = oo or P [log hQ] = - ~ , (1.2) holds trivially.

If both values are finite the strict concavity of x -~ log x implies for P + Q:

P [log hQ] -- P [log he] < log P [hQ/he]

= log/~ [h e Xix ~x: hp (x)> 0}] ~ log /~ [he] = O.

For later applications we shall need a somewhat more general framework. The family of p-measures ~ will be embedded in a family ~ of finite measures. If to each Q e ~ an d -measu rab le nonnegative function h e with # [he] < 1 is assigned such that for all Pc~3, h e is a density of PI~¢ with respect to/~1~¢, then the family f o : = - l o g h e, Q e ~ , is a family of contrast functions for ~]3, if

- o o < P [ l o g he] for all P e ~ and P [ log he] < oo for all P c ~ , Q c ~ , with PW-Q. (This follows as above.)

(1.4) Example: Denote by IR the set of real numbers. Let X =IR and sO= Borel algebra. Assume that for each P e ~ the median r (P) is uniquely determined.

If the 1 ~t moment exists for all P e ]~, the function t ~ ~ I x - t l P(dx) is mini- mized exactly by those values of t which are medians of P. As the median is assumed to be uniquely determined, this implies

~ l x - z ( P ) l P ( d x ) < ~ l x - t l P ( d x ) for all t4=r(P)

and all P c $ . Hence f , ( x ) : = l x - t ] , telR, defines a family of contrast functions with T=IR.

(1.5) Example: Let X = F , , o~/=Borel algebra. Assume that for each Pe~3 the second moment exists. Let z (P) be the expectation of P. We have

( x - z ( P ) ) 2 P ( d x ) < ~ ( x - t ) 2 P(dx) for all telR, t4:r(P),

and all P c $ . Hence f ( x ) : = ( x - t ) 2, telR, defines a family of contrast functions

with T=F , .

z) This means that not P[.f-] =P[f+] = oo.

On the Measurab i l i ty and Consis tency of M i n i m u m Cont ras t Es t imates 251

Denote by IN the set of natural numbers. Let X N be the countable cartesian product of identical components and P ~ [ d ~ the independent product of a countable number of identical components PI~¢. The elements of X ~ will be denoted by _x or (x , ) ,~ . Let, furthermore, a (J//) be the a-algebra over T generated

by ~.

(1.6) Definitions: A s t r i c t e s t i m a t e for the sample size n is an ~¢~, a(ql)- measurable map q~,: X ~ z(~3) which depends on x 1, . . . , x , only. We remark that the word s t r i c t refers to the fact that only elements of r(~3) are admitted as values of the estimate. A m.c. ( m i n i m u m c o n t r a s t ) estimate for the sample size n is a strict estimate for which

(1.7) - - f~.~s)(xi)=inf f(xi) : ter(¢~) . n i=1 i = l

We remark that m.c. estimates exist only under restrictive conditions (because the infimum will not necessarily be assumed on r (¢$)). Therefore we define a somewhat weaker concept:

A sequence of a.m.c. ( a s y m p t o t i c m i n i m u m c o n t r a s t ) estimates (¢p.).~

is a sequence o f strict estimates fi~r which

(1.8) !im exp ,~Lo,,,(x,)-inf exp ~=j;(xi)]'t~z(~) : 0 .

[A more natural definition might seem

,1 , }) lim ~ .. ~ f~.(~)(xi)- inf ~ - - 2 f (xl): t e ~ (~) = O. neN\rl i=1 [ n i=1

This definition is, however, too restrictive, because it does not cover cases in which

inf_ 1 ~ Ji(Xi): te75(~) =--(30 [ n i=1

for infinitely many n~IN. Therefore we use the exponential function.]

Obviously, if ~0, is a m.c. estimate for any nelN, then (~o,),~ N is a sequence of asymptotic m.c. estimates. If we choose f = - l o g h, (see Example(l.3)) the (asymptotic) m.c. estimates become (asymptotic) m.l. estimates.

A sequence (qg.).~ N of estimates is s t r o n g l y c o n s i s t e n t for P if (q~.(S)).~--+ r(P) P~-a.e. I t is strongly consistent for ~3 if it is strongly consistent for any P ~ .

Sufficient conditions for the strong consistency of a sequence of a.m.c, estimates (which also yields strong consistency of a sequence of m.c. estimates) are given in Theorem (1.12). These conditions correspond to the classical conditions for consistency of m.l. estimates. Conditions of a somewhat different nature are given by HtmEg.

252 J. PFANZAGL

We shall need the following notions:

A partially ordered set A is a complete lattice if for any subset A o c A there exist elements in fAoeA and sup Ao~A such that

b < i n f A o i f f b < a for all a e A o

b > s u p A o i f f b > a f o r a l l a e A o.

Let t ~ a t be a map of a topological, space (T, ~ ) into a complete lattice. Then we shall use the following notations:

inf as:= inf {at: teS}

sup as: = sup {at: teS} for any set S c T.

lim a~:= sup infa U, ~ t t ~ U ~ q l

lim a~:= inf sup a v, z ~ t t ~ U ~ q l

{~ s. c. (= lower semicontinuous)

t ~ a, is s.c. (= upper semicontinuous)

lira a~ = at ] iff ~Z' / for all te T.

lim a~ = a t

We shall write fe.~Cto express in a convenient way that f is N-measurable .

(1.9) Theorem: Let (T, ql) be a locally compact HAUSDORFF space with countable

base and let ft: X --* [ - ~ , + ~ ] , t ~ T, be such that

(0) for all n e N and all (xl . . . . . x . ) eX" inf - - ~ ft(xi): te r is attained in T, [ n i = 1

(1) t ~ ft(x ) is l.s.c, for all x ~ X ,

(2) i n f f c C d for all compact sets C c T.

Then for any n EN there exists an ~" , a (o?l)-measurable function tp,: X" ~ T such that

1 f~,~)(xi)=inf ~f~(xl): t e r . 11 i = l i = 1

We remark that condition (2) is always fulfilled if instead of (1) the stronger condition "lim f~= f ]or all te T'" holds.

S ~ t

Proof: According to Lemma (3.8) assumptions (1) and (2) of Theorem (3.10) are fulfilled for the functions

n

(xl . . . . . x.) 7 Z f,(x,), i = 1

t eT .

Hence Theorem (3.10) implies the assertion.

On the Measurability and Consistency of Minimum Contrast Estimates 253

We remark that Theorem (1.9) guarantees the existence of a sequence of m.c. estimates in two important cases:

(1.10) Corollary: Let (T, ~ll) be a compact metrizable space such that conditions (1) and (2) of Theorem (1.9) are fulfilled. I f z (~)= T, then there exists a sequence of m. c. estimates.

Proof: Condition (1.9)(0) follows immediately from Lemma (3.8)(1') and the fact that a 1. s. c. function attains its infimum on a compact set [HAHN, p. 229, 36.23].

(1.1 1) Corollary: Let (T, ql) be a locally compact HAUSDORFF space with countable base such that conditions (1) and (2) of Theorem (1.9) are fulfilled and such that, furthermore, to any xEX, relR, there exists a compact set Cx,rc T such that inf.fcx.r (x) > r (i. e. : f~ (x) tends to + ~ as t tends to "infinity ").

I f z (~) = T, then there exists a sequence of m. c. estimates.

Proof: We have to verify condition (1.9)(0). For a given n-tuple (x 1 . . . . , x,) the function

1

t-~--n i=~=, f~(xi)

is 1.s.c. and tends to + oo if t tends to infinity. Hence it attains its infimum in T.

(1.12) Theorem: Let (T,~ll) be a compact metrizable space and ~31~¢ be a family of p-measures which is parametrized by r: ~ ~ T.

Let ft: X ~ [ - ~ , + ~] , teT, be a family of functions fulfilling assumptions (1) and (2) of Theorem (1.9).

A sequence of a.m.c, estimates is strongly consistent for any P e ~ such that

(3) P [inf fc ] > - ~ for any compact set C c T with z (P)¢ C.

(4) P[f,(e)]<P[f~] for all t~T with t*z(P) .

Proof: Let Poe ~ be a p-measure fulfilling (3) and (4). For notational convenience let to:=r(Po) and To.'=z(¢~).

(i) The strong law of large numbers and the relation

tl n . 1 1 mr - - E f t (xl)- n ~=lfto(xi)

h i = 1 .= t e T b t

imply for PoN-a.a. _xeXN:

1 " 1 " lim inf ± ~ f,(x,)<lim Y'. £o (x,) = Po [ j J • h e n t sTo n i = 1 - - h e n n i = 1

Together with

l i m ( e x p [ - - ~ f ~ ' , ( x ' ( x ' ) l - , i n,=l 4 / = 0 , , ~ \ t_ni=, - _J inrfoeXp ~ - ~ ji(xl)~'~

254 J. PFANZAGL

we obtain for any sequence ((p.).~ ~ of a. m.c. estimates

• 1 n

( 1 . 1 3 ) hm~-~f~.(x)(Xi)<Po[f,o] for Po~-a.a._xeX ~. n~ tl i = l -

We remark that this relation also holds if Po [fo] = - c~.

(ii) Let U ~ °g o be a neighborhood of to. Then the assumptions of Lemma (3.11) are fulfilled for T - U instead of T. Because a 1. s.c. function attains its infimum on a compact set, we obtain from (1.12) (4), (3.11) (a) and (3.11) (b)

(1.14) P o [ f j < infPo[Jl]-<lim inf f (x l ) for Po~-a.a. x e X ~. T - U n ~ N T - U n i = i

(iii) (p,(~)¢ U implies

. 1 " 1 " l n f - - 2 ft(x,)<=-- ~ f~.(x)(Xl).

T - - U n i = 1 h i = 1 -

If there exists x e X ~ such that (p,(s)¢U for infinitely many n, say N 0, then

" l " lim i n f l ~ f ~ (x l )=< lim inf ~,ft(xl) n E I q T - U n i = 1 nERqo T - U n i = 1

(1.15) ai -< lim .,~)(xi)__< lim f~.(~)(xi). ne l ' qo n i = 1 h e n n i = 1

(iv) As relation 0.15) contradicts relations (1.13) and (1.14), there exists a Po ~ - null set M v such that _xq~Mv implies <p,(~)eU for eventually all neIN.

Let ://oC:// be a countable local base of t o and let Mo:= ~ {My: U¢~o}. Then Po N (Mo)= 0 and _x ¢ M o implies for all open neighborhoods U of to:<p . (x)e U for eventually all nelN.

We remark that the compactness assumed for (T,~/) cannot be omitted without compensation. The same holds true for the assumption Po [inffc] > - c~. This follows from Example (2.10) where the corresponding questions are answered negatively for the particular case of contrast functions derived from densities.

The following corollary shows how the compactness assumption can be substituted by other assumptions:

(1.16) Corollary: Let (T, oil) be a locally compact HAUSDORFF space with countable base such that conditions (1) and (2) of Theorem (1.9) are fulfilled and such that, furthermore, f (x) tends to + ~ as t tends to "infinity".

Then any sequence of a.m.c, estimates is strongly consistent for any P e ~ julfilling condition (1.12) (4) and (1.12) (3) for all closed (rather than compact) sets.

Proof: Let (T., o#.) be the one-point compactification of (T, o//). We extend the family f , teT, to T. by defining ft. = - @, where t . is the point at infinity. We have


to show that under the assumptions of the Corollary conditions (1.12) ( 1 ) - (1.12) (4) are fulfilled for (T,,°g,).

(1.12)(1)* trivial.

(1.12)(2)* By (1.12)(2) and Lemma(3.7) we have i n f f v e ~ / f o r all UE~//whence inffveS~¢ for all Ueq/ , . Since t -~ f (x ) is 1.s.c. on T,, Corollary(3.6) implies i n f f c e ~ for all compact sets C c T,.

(1.12) (3)* As the compact sets in T, are the compact sets in (T, ~//) and the sets C u {t,}, C closed in (T, ~//), (1.12)(3) for all closed sets implies P[infJc] > - oo for any compact set C c T, with z(P)¢ C.

(1.12) (4)* trivial.

2. Maximum Likelihood Estimates

In this section we shall apply the results obtained in the Theorems (1.9) and (1.12) to the case of maximum likelihood estimates. Let ~l [.~' be a family of p- measures dominated by a a-finite measure pls¢'. For any P~RI let lip be the equivalence class of densities of PI.~' with respect to P[~. As P + Q implies P [log lip] > P [log lio] for all P e ~13 with P [log lip] > - oo and for all Q ~ ~3, Q + P, with P [log liQ] < oo, any parametrization for which the theory of m.c. estimates is applicable is necessarily 1 -1 . It is therefore natural to assume that ~ is a subset of the parameter space T and z the identity map. For t~T-~3 let an equivalence class liteL 1 (/a) with ~ [lit] < 1 and lit> (3 be given. Then we know from Example (1.3) that - l o g h,, teT, is a family of contrast functions for any choice O~ htElit, te T, i.e.

(2.1) P [log he] > P [log h,] for all (R t)e~13 x Twith P+-t.

To make the results of the preceding section applicable to this case we have to assume that there exist versions in lit, te T, for which assumptions (1.9)(1) and (1.9) (2) are fulfilled. That (1.9) (1) and (1.9) (2) are, in fact, assumptions on particular versions and not on the whole equivalence classes is obvious. That the choice of the versions really influences existence and consistency of m.l. estimates may be seen from the following examples, concerning the particular case T= ~.

(2.2) Example: Let X=[0 ,2) , ~¢ the pertaining Borel algebra and 2[s4 the restriction of the Lebesgue measure. Let ~13={Pt[d: tel1,2)} be a family of p-measures given by their densities with respect to 21~':

1 1 ht:=T.~[o,t] o r h't:=T~[o,t ) t~[1, 2).

For the densities ht, t ~ [1, 2), the function (x~),~ N ~ Pmax(L ~, ....... )is a m. 1. estimate for the sample size n. For the densities h I, re [ l , 2), m.l. estimates do not exist.

256 J. PFANZAGL

(2.3) Example 3): Let (X, ~¢, 2) be as in Example (2.2). Let (a.),~ ~ be a sequence of real numbers with the following propert ies:

(i) 0 < a , < a , + 1 for all n ~ N .

(ii) l i m a , = 1. n---) oo

(iii) ~, a~ < ~ . n = l

An example of such a sequence is a 1 = 0, a z = 3-1, a , = n - z/. for n > 2.

Let f : [0, 1) ~ IR be an increasing function such that f(a,,) = 2" and define for t~[1, 2] a density h'/ of P,/,~ with respect to 2/~/ by:

h~'(x):= f ( t - 1 ) x = t | for t e [1 ,2 )

~0 t < x < 2 J

and h'~(x) - 1 . F u r t h e r m o r e let Po]~¢ be defined by its density h~:=Ztl, 2).

Let $ = { P t : t ~{0}w [1,2]}. Obviously, $ is compac t with respect to the s u p r e m u m metric.

We have hmaxt . . . . . . . . . ) (x l )>2 -1 for i = 1 . . . . ,n and

hm.x(x, ..... x .)(max(xl . . . . . x.))=f(max(x~ . . . . . x . ) - 1), if max(x~ . . . . . x . ) > 1.

Since

Po" {(xl . . . . . x.)61R": max (x 1 . . . . . x,) < 1 + a.} = (Po {x ~ R : x < 1 + a,})" = a,"

and ~ a~ < @,

n = l

the L e m m a of BOREL-CAYrELU implies that for PoN-a. a. _x ~ X ~ we have

max (Xl, . . . , x,) > 1 + a ,

for all sufficiently large n. As condi t ions (1.9)(1) and (1.9)(2) are fulfilled for - l o g h',', t~{0} u [1, 2], we obta in f rom Corol la ry (1.10) that there exists a sequence of measurab le m. 1. estimates, say ( /3) ,~ .

As max (xl, . . . , x,) > 1 + a, implies for all t ~ {0} u [ 1, 2], t 4: max (x I . . . . . x,)

l~I hmax ( . . . . . . . . . ) (xi) i = 1

> 2 - ( " - l ) f ( a . ) > 1, n

Iq h;'(xi) i = 1

we have for Po~-a.a. _x~X~: ~ ( x ) = Pm,x(~ .. . . . . . ) f o r all sufficiently large n. Hence h~,(_~) converges pointwise to h~ (instead of h~) for Po~-a.a. x e X ~.

3) A different example demonstrating that the choice of the densities influences the consistency of the m.l. estimate is given by LANDERS p. 22.


These examples demonstrate that a successful general theory of m. 1. estimates cannot be concerned with m.1. estimates derived from arbitrary versions of the densities. It has rather to assume that the m.1. estimates are derived from distinct versions in ht, teT, fulfilling some regularity conditions. One such useful set of regularity conditions is (1.9)(1) and (1.9)(2). Formulated for m.l. estimates, these conditions are

(1') lim h~=ht for all toT.

(2') sup h c ~ ~ for all compact sets C ~ T. If these conditions are fulfilled, the existence of measurable m.1. estimates

follows under suitable compactness assumptions from Corollary(l.10) or (1.11) (with ~3= T). The most general result concerning the existence of measurable m.1. estimates known to the author is that given by SCHMETTERER, [p. 375, Lemma 3.3] 4). There T is a compact subset of an Euclidean space and t --, h, (x) is assumed to be continuous (with respect to the usual topology of the Euclidean space T) for all x E X . The result obtained from Corollary (1.10) is more general in two respects:

a) T is an arbitrary compact metric space, b) continuity of t ~ h t (x) for all x e X is substituted by u. s. c. In connection with b) we shall mention that there is a general class of families

of p-measures, including the monotone likelihood ratio-families, admitting densities which are u.s.c, with respect to the sup-metric in ~, for which it is, however, not possible in general to choose versions of the densities which are continuous everywhere (see PFANZA6L). A particular instance of such a family is given in Example (2.2).

The following Criterion (2.4) gives conditions under which versions of densities fulfilling (1') and (2') exist (for T = ~). These conditions are expressed in terms of the equivalence classes of densities. The author considers this essential, because continuity requirements expressed in terms of particular versions of the densities depend on the whole family of p-measures, whereas continuity requirements expressed in terms of equivalence classes of densities can be verified by showing them for one set of arbitrarily chosen versions. Furthermore it suffices to prove u. s.c. sequential continuity (see Lemma 3.3).

It was remarked by WaLt) that Condition (2') can always be fulfilled by choosing a separable version of the densities (separability in the sense of DooB [-pp. 52]). The criterion below gives sufficient conditions under which even "more regular" versions of the densities can be selected. Such more regular versions are needed, because families of versions with properties(1.9)(1) and (1.9) (2) retain these properties for arbitrary sample sizes (whereas it seems doubt- ful whether separability carries over to arbitrary sample sizes).

4) The proof of this Lemma is attributed by SCHMETTERER to KRICKEBERG. It is, in fact, already contained in a coded form in RICHTER 1-1963, p. 88, Hilfssatz 4]. Other relevant references are YON NEUMANN [1949, Lemma 5, p. 448] and Slow [1960, Theorem 4.1, p. 238].

258 J. PFANZAGL

(2.4) Criterion: Let ~ 1 ~ be dominated by a a-finite measure ~1~/ and endowed with a metrizable topology with countable base. For any Pc~3 let he be the equivalence class of densities of P I d with respect to l*ls¢. I f

(2.5) ~moho=h P for all P ~ 3 ,

then there exists a system of densities hpehp, Pe~3, fulfilling (1') and (2') (with T= ~3). Conversely, if there exists a system of densities fulfilling (1'), this implies (2.5).

Proof: From (2.5) we obtain by Theorem (3.4), applied for - h e , P ~ , that there exist versions h e ~ e , Pe~3, he>O, such that

lim h e = h e for all Pc Q ~ P

and sup hvesu p hv for all open sets U c:. ~ . This proves (1') and implies by Corol- lary (3.6) that sup h c e ~ for all compact sets C c ~. This proves (2'). Conversely, (1') implies (2.5) by Lemma (3.3).

Sufficient conditions for the strong consistency of a.m.1, estimates are immediately obtained from Theorem (1.12) and Corollary (1.16).

(2.6) Theorem: Let T ~ 3 be endowed with a compact metrizable topology. For any tE T let h,~h t be such that (1') and (2') are fulfilled.

A sequence ofa.m. 1. estimates is strongly consistent for any P E ~ such that

(3') P [log sup hc] < ~ for any compact set C c T with PC C.

(4') P [log he] > - oo.

Proof: Follows immediately from Theorem (1.12) and the fact that - l o g hi, te T, is a family of contrast functions (see (2.1)).

We remark that condition (2') is always fulfilled if instead of condition (1') the stronger condition "lim hs= h, for all t~ T" holds.

S ~ t

(2.7) Criterion: Let T ~ 3 be a locally compact HAUSDORF space with countable base. I f

(1 +) lim h~= h, .[or all t e T, S ~ t

and for some Po~ ~3

(3+) Po [log sup hc] < oo for any compact set C c T with 19o¢ C,

(4+) Po [log heo] > - ~ ,

then there exist versions h , ~ , , t~ T, fulfilling the conditions of Theorem (2.6) with P = Po. Conversely, if there exists a system of versions fulfilling the conditions of Theorem (2.6) (for some Po ~ ~ ) this implies (2.7) (1 +), (3 +), (4 +).

Proof: By (1 +) we may choose (Theorem (3.4)) versions h,~ti t, t~ T, such that (1') holds and sup hv~suph v for all open sets U c T . Corollary(3.6) implies sup hc~a¢ for all compact sets C c T. It remains to show that these versions fulfill (3'). Let C be a compact set with Poe C. As T is a locally compact HAUS- D O R F F space, there exist a compact set C~ and an open set U such that t¢ C~ ~ U ~ C.

O n t h e M e a s u r a b i l i t y a n d C o n s i s t e n c y o f M i n i m u m C o n t r a s t E s t i m a t e s 259

Therefore sup h c < sup h vesup hv < sup hcl.

As by (3 +) P0 [-log sup ]1c~] < o% we conclude that Po I-log sup hc] < oo. This proves the first part of our statement. The converse follows immediately from Lemma (3.3) and the fact that sup h c < (sup hc) for all compact sets C c T. (2.8) Corollary: Let TD~3 be endowed with a locally compact HAUSDORFF topology with countable base. For any t ~ T let hteh t be such that (1') and (2') are fulfilled and h, --* 0 if t tends to the point at infinity.

A sequence of a. m. I. estimates is strongly consistent for any P ~ ~3 which fulfills (2.6) (4') and (2.6) (3') for all closed (instead of compact) sets.

Proof: Follows immediately from Corollary (1.16) and the fact that - l o g ht, t~ T, is a family of contrast functions (see (2.1)).

WALD and in his succession also SCHMETTERER [p. 379, Satz 3.6] uses in the case T = ~ instead of (2.6) (3') the following seemingly weaker condition: To any P + P0 there exists a real number r(P) such that Po I-log + sup hnp] < oo for

Bp.. = { Q e ~ - d(Q, P)<r(P)}.

In fact, this condition is equivalent to (2.6)(3') (if T = ~13). For, let C = ~3 be an arbitrary compact set, not containing Po- Then {B~: P c C} is an open cover of C and hence contains a finite subcover, say B~,, .. B ° • , Pk [ B° denotes the interior of B]. Then, suph c< max sup hap.

i=1 , . . . ,k This, however, implies

k

log sup hc< log + max sup hnp = max log + sup hBp < ~ log + sup hap ,. i = 1 . . . . . k ~ i=1 , . . . , k i i = 1

Hence P0 [-log sup hc] < co.

The following examples demonstrate that neither the compactness condition nor any one of the conditions (1'), (3'), (4') of Theorem (2.6) can be omitted without compensation. More precisely: If at least one of these conditions is violated, then the m.l. estimates may fail to be consistent.

The first example (due to LANDERS p. 30) shows that neither the compactness condition nor the continuity condition can be omitted without compensation.

(2.9) Example: Let X = [ 0 , 1) and d the pertaining Borelalgebra. For each

(°) 2 < m e N , we define p-measures in the following way: For any set m

{k~ . . . . , kin} ~ {1 . . . . , m 2 } ,

P~k ...... k~) is defined by its density with respect to the LEBESGUE measure'

2 for xe U F k i - I ki ~=,L ~ 'm ~

htk ...... k~)(X)= m--2 elsewhere. m - 1

260 J. PFANZAGL

Let ~ consist of the LEBESGUE measure 2/~1 and all m e a s u r e s P(k .. . . . . kin) with {k~ . . . . , km} c { 1 . . . . . m2}, 2 < meN. As 2 is finite and as the densities are uniformly bounded, conditions (2.6)(3) and (2.6)(4') are fulfilled. Furthermore,

d(R, Plk ...... k ~ , ) = m -1

which implies that ~ is compact with respect to the supremum metric. The continuity condition (2.6)(r) is, however, not fulfilled for the limiting measure 2. (We remark that d (P, Q).'= sup {tP(A)- Q(A)I: A ed}. )

We shall show that any sequence of m. 1. estimates converges to 2 (with respect to the supremum metric) with probability 1 for any measure of ~ (and is therefore not consistent except for 2). As d(2, Ptk ~ ..... k,o)=m-x, it suffices to show that ( m ( x l , . . . , x~)),,~N diverges to ~ for almost all _x~X ~, where m ( x l , . . . , x~) is the order m of the tuple (k I . . . . . kin) assigned to the m. 1. estimate.

If ( m ( x 1, . . . , x , , ) ) ~ N o remains bounded for some subsequence No-IN, then there exists mo~N, m o >2, such that at least one of the mo-tuples assigned to the sequence of m.1. estimates, say (k°,... o , k~o), occurs infinitely often. As

and as

n

[ [ htk, ..... k~ (X , ) < i = 1

n

l-I htk ...... k,)(Xi) = i = 1

2" for all m-tuples (kl . . . . , kin), 2 < m e N ,

2 ~ for some n-tuple (kl, ..., k~)

containing all the integers [/,/2 Xi], i=-1,..., r/ [where [r] is the smallest integer greater than r], the value 2 ~ is also assumed for the m. 1.-estimate. This implies that

n

I-I htk ° ..... k°..o ~ (Xi) = 2 ~ i = 1

for infinitely many neN. Hence

m o

x i e A (k o, o .... k..o): = U [( k ° - 1)/mo 2; k ° / m 2 ) i = 1

for i = l , . . . , n

for infinitely many n e N and therefore x i ~ A (k°, ..., k°o) for all ieN. As

2 P ( A ( k° . . . . , '~,,,,,v . = for all P e ~ ,

mo

the set of all ( x~ ) i~NeX ~ with x i e A ( k °, o . . . , kmo ) for all i eN is a PN-null set for all Pe~ ,na tu r a l l y depending on (k °, o . . . . kmo ). As the set of all m-tuples, meN, is countable, this implies that the set of all ( x ~ ) l ~ e X N for which ( m ( x I . . . . , x,))n~N °

is bounded for some subsequence N o c N has PN-measure zero for all P e ~ . Therefore for all P e ~ the sequence ( m ( x 1 . . . . , x~))n~N diverges to oo P~-a.e.


If we consider the family ~3 - {2}, then the continuity assumption with respect to the supremum metric is fulfilled in a trivial way because m + n implies

d(P(k ...... kin), PU ...... l,))>= I m-x -- n- l l ,

whence any element of ~ - { 2 } is isolated. Assumptions(2.6)(3') and (2.6)(4') remain valid, too, and it is now the lack of compactness which is responsible for the unpleasant behavior of the m. 1. estimates.

The following example shows that conditions (2.6)(3') and (2.6)(4') cannot be omitted without compensation 5).

(2.10) Example: Let ~ be the family defined in Example(2.3). This family is compact with respect to the supremum metric and for the family of densities h;', te {0} w [1, 2], only assumption (2.6)(3') of Theorem(2.6) is violated for t=0 . The m.1. estimates are not consistent for t = 0. If the dominating measure 2/,~ is replaced by the measure/1/~4, given by its 2-density

1 xe[0, 1) g(x)= f ( x - 1) xs[1, 2)

then the family of p-densities given by h','/g, re{0} u [1, 2], fulfills (2.6)(1')-(3') but not (2.6) (4'); m.1. estimates for this family of densities are also m.l. estimates for h~', te{0} t_; [1, 2] whence they are not consistent for t=0 . We remark that the family ~ in Example (2.3) fulfills (2.7)(1+), (3 +), (4 +). Hence there exist versions of densities, fulfilling (2.6)(1')-(4') (e.g. the densities h t specified in Example (2.2)).

Corollary (2.8) demonstrates one possibility to substitute the compactness condition of Theorem (2.6) by other conditions. If X itself is a locally compact HAUSDORFF space with countable base we can use the fact that the family Ello;¢ of all measures Q l d with Q ( X ) < 1 is compact in its vague topology [see BAUER, 1968, p. 193, Korollar46.3]. We remark that the vague topology is defined by the following local subbase at Qo ~ El: {Q ~ El: [Q I f ] - Qo I f ] [ < ~}, e > 0, f a real valued function with compact support.

Let ~ be a given family of p-measures and ~c its vague closure in El. (We remark that the elements of ~3 c - ~ are not necessarily p-measures any more.) As ~c is compact with respect to the vague topology, and as the vague topology is metrizable under the assumptions made for X, [see BAUER, 1968, p. 194, Satz 46.4], we may apply Theorem (2.6) for T= ~c and q /= vague topology, if conditions (1') and (2') are fulfilled on ~ . Then a sequence ofa. m. 1. estimates is strongly consistent for any P e ~ such that

(3") P [log sup hc] < oo for any compact set C c ~ with P ¢ C.

(4") P [log hp] > - oo.

5) Other examples with this effect were given by BAHADUR [p. 208] and LANDERS [p. 31]. See also example 2.15 which is, however, less transparent.

262 J. PFANZAGL

We remark that for all Q ~ ~ c _ ~ , the functions hQ are restricted by p [hQ] < 1 (see (2.1)).

A natural condition on ~ which guarantees that (1') and (2') are fulfilled on ~c with suitable functions hQ, Q ~ ~ c _ ~ is:

(2.11) There exist densities he, P ~ , such that P ~ h p ( x ) is uniformly continuous on ~ for all x ~ X.

Then for any x ~ X P ~ h p ( x ) , P e ~ , has an unique (uniformly) continuous extension to ~c [DUNFORD-ScHWARTZ, p. 23, Theorem 17]. We remark that hQ is not necessarily a density of Q 1~¢ with respect to p I~¢ if Q ~ ~ c - ~ .

For any Q ~ ~ c _ ~ there exists a sequence (P~),~ ~ c ~ , converging to Q (in the vague topology). Hence

lim hp.(x)=ho(x ) for all x ~ X n~N

and the Lemma of FATOU implies

/~ [hQ] = # [!!m h,.] < ~ # [-h,.] = 1.

Consequently - l o g he, Q ~ ~c, is a family of contrast functions for ~13 (see (2.1)), fulfilling (1') and (2'), because Q ~ he(x), Q ~ 3 c, is continuous for all x ~ X . Thus we obtain

(2.12) Theorem: Let (X, ~ll) be a locally compact HAUSDORFF space with countable base, s l the pertaining Borel algebra, and ~3 I d a family of p-measures fulfilling condition (2.11).

A sequence of a.m.I, estimates is strongly consistent for any P ~ 3 such that

(3') P [log sup hc] < oo for any vaguely compact set C ~ ~3 ~ with P ~ C.

(4') P [log he] > - oo.

If X = N , the assumptions of Theorem (2.12) are fulfilled for the discrete topology. As convergence of p-measures with respect to the vague topology is in this case equivalent to convergence of p-measures with respect to the sup- metric, condition (2.11) is in this case always fulfilled. If we take for the dominating measure p the counting measure, we have hp(x)< 1 for all x eX , P e ~3 c, whence log sup hc(x)<O for all x e X and any compact set C c ~ c. Hence condition (3') is always fulfilled. Thus we obtain the following well known result [see KIEFER and WOLFOWITZ, p. 893]

(2.13) Theorem: / f ~ is an arbitrary family of p-measures over N, a sequence of a. m. l. estimates is strongly consistent for any P ~ ~ such that

(2.14) ~ P{k} logP{k} > - o o (where P{k} log P {k} = 0 if P{k} =0). k = l

The following example shows that m.1. estimates may fail to be consistent if condition (2.14) is violated. Another - slightly more complex - example with


the same effect was given by BAHADUR [p. 208]. This example gives a countable family of p-measures such that the m.I. estimates are failing to be consistent for all but one p-measure. In the example given below, condition (2.14) is violated for two p-measures of the family: Po and Po~. The m.1. estimates are consistent for all p-measures except Po.

(2.15) Example: Let (a,),~N be an increasing sequence of positive numbers such that oo

l i m a , = l and ~a , "<oo , n~cJo n=l

(see Example (2.3)). For any heN, let h be the largest integer l such that 2~<n.

We define p-measures Po, P~, P~, meN, as follows:

0 for k=O Po{k}= 2_~(a~+l_a~) for k e N ; where ao :=0

{22-' for k = 0 Po~ {k} = ~ Po {k} for k e n

[2-1(1 +a,~-a,~+l) for k = 0

"k" [2-1P°(k) for l < k < 2 a+l, k+-m P~/ ' = / 2 _ , Po(k)+2-t (1-aa , ) for k=m

[0 for k > 2 '~+1.

The family {Po,P~, Pro, meN} is dominated by the measure #{k}= 1 for k= 0, 1, 2 . . . . and compact with respect to the supremum metric. Nevertheless, if Po is the true p-measure, any sequence of m.l. estimates converges to P~ with probability 1. This is possible, because assumption (2.14) is not fulfilled for Po.

It is easy to see that P~ {k} > 2-1Po {k} for k = I ..... m - 1 and P,, {m} > 2 'h -1Po {m}. Hence

h n Pmax( . . . . . . . . . }{Xl}>2-.+, 2{max( . . . . . . . . . )V-1H Po { x i } . i = 1 i=1

A s 2--1

Po {k} =a . , k = l

we have Po~ {(xj)j~ ~: max(xl . . . . . x,)_<2 ~-- 1}-- a,". As

Z a."<oo, n = l

the Lemma of BOREL-CANTELLI implies that PoN-a. e. for all sufficiently large n E N: n n

(2.16) ~ Pmax( ......... ){X,} > I]Po {xi}. i=1 i = l

Let ( / ~ ( X 1 . . . . . Xn))n~l N be the sequence of indices corresponding to a sequence of m. 1. estimates. By (2.16) we have rh (xl, ..., x . )# 0 PoN-a. e. for all sufficiently large n.

264 J. PFANZAGL

Furthermore,

12IP~{x/}=0 if meN and max(x1 . . . . . x~)>2 ~+1. i=1

Hence vh (x 1 . . . . . x~) > 2-1 max (x , ..., x,). As (max (x I . . . . . x . ) ) ~ converges to infinity Po~-a.e., (rh(xx . . . . , x , ) ) ,~ converges to infinity Po~-a.e., too. Therefore, the sequence of m.1. estimates, (P,~t ......... ~),~, converges to Po~ with respect to the sup-metric Po~-a. e.

3. Auxiliary Lemmas and Theorems

To make the paper more legible, all auxiliary considerations which are not relevant for an understanding of the main results (presented in sections 1 and 2) are collected in this section. Some of them, e.g. Lemma (3.8), Theorem (3.10) and Theorem (3.4), may have some interest of their own.

Given a measure space (X, ~ , #) with a-finite measure p, L denotes the system of all #-equivalence classes of d-measurable functions f : X ~ [ - ~ , + Go]. The elements of L will be denoted by f

(3.1) Lemma: (L, <=) is a complete lattice, i.e. to any family J~e L, t 6T, there exist elements i n f ideL and s u p j ~ L such that for all ~,~L

infj~>__~ iff ~>=~ for all t~T

supfr=<~ iff £<=~ for all teT.

Furthermore there exists a countable subset T oc T such that infJ~o=infJ~ [see DUNFORD-SCHWARTZ, IV.11., Theorem 6, p. 335].

The importance of the relation infJ~ o = i n f ~ lies in the fact that the infimum over countable families {£: teTo} can be taken pointwise. More precisely: For any family f r e d , t e T o, with fteft for all teTo, we have inffToeinffr o (whereas this relation is not true, any more, for uncountable families). (3.2) Corollary: To any family f teL, t~T, and any countable family ~ll o of subsets of T there exists a countable set T O c T such that infJ~nr o = infJ~ for all U ~g~ (where ~11~) denotes the smallest system of sets containing °1lo which is closed under countable unions).

Proof: According to Lemma (3.1) to any U~//o there exists a countable set T v c U such that infJ~v=infJ~. For To:=(.. ) {Tv: Ueq/o}, we have infJ~nTo= infJ~ for all Ue~//o . Let U~q/~. Then there exist Uned//o, n~IN, such that

u= U on. n ~ N

We have

inffu = i n f f u v. = inf infiX. = inf infJ~.~To = i n f f u v, nro = infj~ ~ro.


The following Lemma gives sufficient conditions on T and L under which 1.s.c. is equivalent to lower sequential semicontinuity:

(3.3) Lemma: Let (T, ~ ) be a HAUSDORFF space fulfilling the first axiom of countability, I~[d a a-finite measure and J~L , t~T, a given family. Then.for any t~T the following assertions are equivalent

(i) lim J~, > f~ for any sequence (tn)~ ~-~ t.

(ii) lim f~= f~.

Proof." (ii)~(i): Follows immediately from

lira f~n > lira f~.

(i) ~(ii): Let t ~ T be arbitrary. As ~// fulfills the first axiom of countability, there exists a decreasing sequence U ~ / / , nelN, which ;s a local base at t. By Corollary(3.2) there exists a countable set T o ~ T such that inffv,~To=infJ~. for all n~N. Furthermore, for any z~T we choose a fixed version f~f~.

Let S,,k consist of the first k elements of U, c~T o and let

A., k: ~--- {X ~ X " infJs., ~ (x ) - infjv" ~o (x) > 1/n}.

Without loss of generality we may assume that p is finite and that f,, te T, is uniformly bounded. Then (A., k)k~r~J, (~ implies

l im~ (A., k) = 0. k ~ o o

Hence to any h e n there exists k.elN such that p(A., k.)< 1In. We define S.: = S., ~. and obtain for all heN:

p {x ~ X: inffs" (x ) - inffv" ~ To (x) > l/n} <= 1In. Let

T~.'= 0 S~. v = n

As S~c U~ c~Toc U~ c~To for all v>n, we obtain S~c T~c U~c~To. Hence for all neN:

g {x eX: inffr ~ (x ) - inffv" ~ To (x) > 1/n} <= l/n.

We have for arbitrary r>O

{xeX: lira inffr " (x ) - l im inffv.~r o (x)> r}

~ ~) ~) {xeX: inffr~(X)-inffv~ro(X)>r}. m = l n = m

Hence /~ {xeX: l im inffT~ (x ) - l im inffu~ ~To (x)> r}

< lim/~ {xeX: inffT. (X)--inffv~nTo(X)> r} =0 , n ~ o o

266 J. PFANZAGL

a s I~{xeX: inffr.(x)-inffv.~ro(X)>r} <l/n for all n>l/r.

As r > 0 was arbitrary, this implies

lim infJ~. = lim infJ~.~To. n ~ o o n ~ o o

Let (t,),,N be a sequence made up of 7"1 such that t ,= t for infinitely many n e N if t e ~ T,. It is easy to see that

riG:IN

lim t, = t. n ~ o o

We have

lim f~. = lim inffr. n ~ c t ) t l ~ c~3

Hence (i) implies

= lim infJ~.~,ro= lim i n f f v = sup in f fv=l im ] ~. n ~ n ~ t E U E ~ Z ~ t

l im~>f~. ~ t

As the converse relation is trivial, this proves (ii).

(3.4) Theorem: Assume that (T°//) has a countable base. Let J~eL, teT, be a family with (*) limf~ = 2 for all teT.

Then there exist ftsJ~, teT, such that

(i) t--*f(x) is l.s.c, for all xeX.

(ii) inf fveinfJ ~ for all ueql.

We remark that if J~<<_O for all t eT (where () is the equivalence class containing the function identical to zero) the functions may be chosen nonpositive.

This theorem is related to DooB's theorem on the existence of equivalent separable random functions: It states in addition that for l.s.c, equivalence classes standard modifications can be chosen which are 1.s.c. everywhere. (We remark that a corresponding theorem with "continuous" instead of "l.s.c." is not true any more.) Moreover, the theorems of DOOB [p. 57, Theorem2.4] and NEVEU [p. 89, Proposition III.4.3] are confined to the special case T = R and assume (X, ~,/~) to be complete. It was remarked by BAUER [1964] and BORaES that these two restrictions can be easily removed.

Proof: Let q/oCq/ be a countable base. According to Corollary3.2 there exists a countable set T o c T such that

i n f J ~ r o = i n f f ~ for all U e ~ o .

As ~o is a base of ~ , we have

sup inffv-= sup inffv. t eUedl lo t ~ U ~

Together with (,), this implies

(**) sup inffv~ro=f~ for all t~T. t~Ue~o


For any t E T o we choose gt~f, and define

X, := { x ~ X : sup inf gwTo(X)~=gt(x)} t~Ue~o

Xo,= Ux,. t~To

(**) implies #(XO=O for all teTo whence #(Xo)=O. Now we define a family f t ~ ¢ , t~T, as follows:

f~:=SZXo.~ gt for tmT o ~ZXo sup i n f g w r o for tCT o.

t~OE~o

We remark that f is ~-measurable as all the functions occuring in the definition are d-measurable. As p(Xo)=0, we immediately obtain f e f~ for t e T o.

Furthermore, (**) implies sup inf gv ~ ro e f

and therefore f~f~ for tCT o. Finally,

(***) f~= sup inffv~T o for all teT. t ~Ue~o

Let (t, U)~Tx~ll with t ~ U be given. Then there exists Uo ~ ~//o so that t~Uo C U.

From (***) we obtain

i n f f w To < inffvo ~ro < ft. Therefore,

inffv = inffv n To e inffwT~ = infiX.

Together with (***) we obtain

J~=limf~. z ~ t

(3.5) Lemma: Let (T,~//) be a metrizable space and let t ~ a, be a I.s.c. map of T into a complete lattice A which is linearly ordered. Then to any compact set C c T

there exists a sequence of open sets U , ~ l , neIN, such that

infa c = sup infau .

Proof: Let C be an arbitrary compact set and let U,.'= {teT: d(C, t )<I/n} . Then for any V~°//with C = V we have C c U,c V for all sufficiently large n e N (because C and V are disjoint and therefore of positive distance).

If b < i n f a c, we have b<a, for all t~C. As t--~a t is l.s.c., to any t e C there exists U~e°g with teUt such that b<infau, . As {Ut: teC} covers C, there exists

a finite subcover, say U~,,..., U~. This implies

U.= 0 U,, i=1

268 J. PFANZAGL

for all sufficiently large n and therefore

supinfav. >infa ~ v , = inf i n f a v , > b . n e N - - i = l ' i = 1 , . . . , m ,

Hence sup inf au, < in fa c would lead to a contradiction. As C c U,, nelN, implies t t ~ I

sup inf au. < inf a c, we conclude that inf a c-- sup. inf a v .

(3.6) Corollary: Let (T, ~ll) be a metrizable space. Let fe~c], te T, be a family such that t ~ ft(x) is l.s.c, for all x e X . Then

i n f f v e d for all U e~l implies i n f f c e ~ for all compact sets C ~ T.

Proof: Follows immediately from (3.5) applied for a,'.=ft(x).

(3.7) Lemma: Let (T, ql) be a a-compact metrizable space and f : X-* [ - ~ , + ~ ], teT, a family of functions. Then i n f f c E ~ for all compact sets C ~ T implies i n f f v e d for all Ueql.

Proof: Let U E ~ be arbitrary. There exists an increasing sequence of compact sets C,, nEN, such that

U = U C.. n ~ N

For any te U we have te C , c U for all sufficiently large n. Hence

ft > lim inf fc, > inf fv . n e ~ I

As t6 U was arbitrary this implies

i n f f v = l i m inffc e .~ . nEh'q n

(3.8) Lemma: Let T be a metric space and ji(i): X--* [ - ~ , ~ ] , t6T, i=1 . . . . . n, a family of functions such that

(1) t ~ ft(i)(x) is l.s.c, for all x ~ X , i= 1, ..., n.

(2) inf fc")~,~ for all compact sets C; i= 1 . . . . . n.

Then

(1') t ~ ~ f(i~(x) is l.s.c, for all x ~ X . i = 1

(2') inf f"~: t e C ex4 for all compact sets C. ti=l

Proof: (1') is well known [HAHN, p. 297; 36.1.51]. We shall show that there exists a sequence of compact sets, say C k, kslN, such that

inf f(i): t e C = inffct~ ). i i = l


This relation then immediately implies (2'). Fo r any m ~ N we determine a finite number of sets

B , , , : = { t ~ C : d(t,t~)< 1 } 1=1 . . . . . L(m)

such that Bin, 1 . . . . , B,,,um) covers C. Let (Ck)k~ ~ be a sequence made up of the sets B,, 3, l = 1 . . . . , L(m); m ~ N . Let t e C and let lNt: = {nelN: te Cn}. We remark that N~ is an infinite set. We have ft~i}>inffc~ ) for all k~IN, whence

~ f,{" > lim inf fc~2 > 2 inf fc~2. i = 1 k ~ N t i = l i = 1

As t ~ C was arbitrary, this implies

inf "~: tEC >l im inf . k i = 1 k ~ q i = 1

To prove the converse relation let x ~ X be arbi t rary and let N x ~ N be an infinite subset such that

(÷) lim ~ in f fc~2(x)= l im inffc~2(x). keNx i= - ~ - 1 " k~N i= 1

To any k~]N and i~{1 . . . . . n} there exists [HAHN, p. 299] ~i, kECk such that ft ") inf fc~)(x). As metric space, it is also sequentially compact. ~. (x)= C is a compact

t X Hence there exists an infinite subset IN x ~IN~ and t~e C such that (t Lk)keNx t~e C. It is easy to see that

lim d ( C k) = O . k ~ o o

As t~,kE C k for i = 1, . . . , n, this implies (tiXk)kerN, -~ t x for all i-- l, . . . , n. Therefore,

n n n n

lim ~ inffc~)(x)= lim ~ inffct: ' (x)= l i m _~lft~)k(X)~> i_~ 1 lim f~ ' (x) k~Nx i = 1 k ~ N x i = 1 k ~ q x i ~ x t , , k

> f ~ ) ( x ) > i n f "~(x): t e C . i = 1 i

Together with (+) this implies

lim i n f f c ~ ( x ) > i n f f~i~(x): t e C for all x e X . k e N i = l i

The following Theorem is essentially due to SION [Theorem 4.1]. The version which is most appropr ia te for our purpose is to be found in LANDERS [p. 10].

(3.9) Theorem: Let (T, oll) be a locally compact HAUSDORFF space with countable base and ~ its Borel-algebra. Let (X, d ) be a measurable space. Let cp: X--* ~ ( T ) be a map 6) which assigns to each x e X a nonempty closed set ¢ ( x ) ~ T such that

6) ~(T) denotes the power set of T.

270 J. PFANZAGL

{x~X: C n ~ ( x ) - - ~ } e d for all compact sets C c T. Then there exists an d , ~ - measurable map tp: X ~ T such that q~ (x)~ ck (x) for all x e X.

(3.10) Theorem: Let (T, #1) be a locally compact HAUSDORFF space with countable base. Let ft" X ~ [ - ~ , + oo1, t e T, be a family of functions such that

(1) t -* ft(x) is l.s.c, for all x e X .

(2) i n f f c ~ d for all compact sets C.

(3) For each x e X there exists t ~ T such that f t (x)=inf f r (x ).

Then there exists an d , a (~)-measurable map ~o : X - * T such that

f~tx~(x)=inf f~(x) for all x ~ X .

Proof: For an arbitrary x e X , let ~(x ) :={ teT: f~(x)=inffT(x)}. By assumption (3), we have ~(x)~:0.

Furthermore, ~(x) is closed: Let t be an accumulation point of ~(x). Then teUe~ll implies U n ~ ( x ) ~ O and hence inffv(x)=infr(X). Using (1) we obtain

£ (x) = sup i n f f v (x) = inf T (x) tEU~,I

and therefore t ~ ( x ) .

Finally we shall show that { x e X : C n ~(x)=~)} e ~ for any compact set C: We have C n • (x) = ~ iff ft (x) > inffT (x) for all t e C iff inffc (x) > inffT (x) [HAHN, p. 299; 36.2.31.

Hence {x eX: C n q~ (x) = ~b} = {x eX: inffc (x) > i n f f r (x)} ~ ¢ , because inffT is ~¢-measurable according to Lemma (3.7).

The existence of tp now follows from Theorem (3.9).

(3.11) Lemma: Let ( T, all) be a a-compact metrizab le space and ft: X ~ [--oo, + oo ], t eT, a family of ~-measurable functions such that:

(1) t ~ f ( x ) is I.s.c. for all x e X .

(2) i n f f c ~ ¢ for any compact set C c T.

Let furthermore Plsg be a p-measure such that

(3) P [ inffr I > - oo.

Then

(a) t ~ P[f t] is l.s.c. 1 "

(b) inf P [ft] < lim i n f - - ~ f (xi) P~-a. e. for any compact set C. t~C n ~ t E c n i=1

The basic idea of this Lemma is due to WALD. The present form is essentially Lemma 5 of LE CAM [p. 3011. As a number of measurability questions is not discussed by LE CAM, we shall give the proof in full detail. For continuous (instead of 1. s. c.) functions and T = ~ x k this lemma is an immediate consequence of Lemma 3.5 of SCHMETTERER [p. 3891.

O n the M e a s u r a b i l i t y a n d C o n s i s t e n c y o f M i n i m u m C o n t r a s t E s t i m a t e s 271

Proof: (a) We remark that by Lemma (3.7) i n f f c ~ for all compact sets C c T implies i n f f v e d for all open sets Ue~//. If r<P I f ] , then there exists Ute~i, with tEU t such that r<P[inf f j . To see this, let V, eq/, n~lN, be a decreasing neighborhood system of t. By (1) we have

fi = lim in f fv . n~c~

Because of (3) the monotone convergence theorem is applicable and yields

lim P [inffv,] = P [ft] > r. n ~

Hence P[infv,]>r for all sufficiently large h e n and we choose one of these V, for U~. As f~>inffv, for all s e U , we have

infP [f~] > P [inffu~] > r. se-Ut

This proves (a).

(b) First of all we remark that for each compact set C c T

: i inf fi(xi) t e C rl i=1

is measurable according to Lemma (3.8). If r < P [ f t ] for all teC, we determine to any te C a neighborhood Urea//as in (a). {Ut: teC} is an open cover of C and therefore contains a finite subcover, say Ut,, .... Utk.

By the strong law of large numbers, to any t6C there exists a PN-null set Z, such that x ¢ Z~ implies

lim 1 £ inffv, (xl) = P [inffv,] > r. n~oo H i = 1

We remark that this relation also holds if P [inffv,] = + ~ . As

k

c Uu,,, i = 1

we have

n n 1 n

i n f - - E f ( x , ) > min inf l ~ f ( x , ) > min - - E i n f f v , (x,) ~C tl i = l - - j = l , . . . , k t~Utj n i j = l , . . . , k Y/ i=1 J

and therefore

1 " 1 " lim inf--~£(xi)>__ min lim--~.inffv,(xl)= rain P[inff%]>r

n ~ - ~ t~C 1l i=1 - - j = t . . . . . k ~ /'/ i =1 J j = l . . . . . k

for

The exceptional P~-null set

k

j = l

k

U z,j j = l

272 J. PFANZAGL

depends on r. The union over the P~-null sets for all rational r, say Z 0, is a P~- null set again. For _xCZo,

infP[f]>r implies lim inf ~1 t ~ C n ~ o v t E C n i = 1

for all rational r whence the assertion follows.

Acknowledgement: The author wishes to thank Mr. P. G)~NSSLER and Mr. D. LANDERS for valuable suggestions. Mr. LANDERS eliminated an error in the proof of Criterion 2.7.

References

BAHADUR, R. R.: Examples of inconsistency of maximum likelihood estimates. Sankhya 20, 1958, 2 0 7 - 210.

BAUER, H . : Markoffsche Prozesse, Vorlesungsausarbeitung, Hamburg, 1964. - Wahrscheinlichkeitstheorie und Grundztige der MaBtheorie, Berlin, 1968. BORGES, R.: Zur Existenz yon separablen stochastischen Prozessen. Z. Wahrscheinlichkeitstheorie,

6, 1966, 125-128. DooB, J. L.: Stochastic Processes, NewYork, 1953. DUNFORD, N., and J. T. SCHWARTZ: Linear Operators, Part I, New York, 1966. HAHN, E.: Reelle Funktionen, New York, 1959. HALMOS, P. R.: Measure Theorie, Princeton, N.J., 1950. HUBEg, P. J.: The behavior of maximum likelihood estimates under non-standard conditions, Proc.

5th Berkeley Symp. on Math. Statist. and Prob., Vol. I, 1967, 221-233. KIEe~R, J., and J. WOLFOWITZ: Consistency of the maximum likelihood estimator in the presence of

infinitely many incidental parameters. Annals Math. Stat. 27, 1956, 887-906.

LANDERS, D.: Existenz und Konsistenz yon Maximum Likelihood Sch~itzern. Thesis, University of

Cologne, 1968. LE CAM, L.: On some asymptotic properties of maximum likelihood estimates and related Bayes'

estimates. University California PubL Statist. 1, 1953, 277-330. voN NEUMANN, J.: On rings of operators. Reduction theory. Ann. of Math. 50, 1949, 401 - 485. NEVEU, J." Mathematical Foundations of the Calculus of Probability. San Francisco-London-Amster-

dam, 1965. PFANZAGL, J.: Further remarks on topology and convergence in some ordered families of distri-

butions. Annals Math. Stat. 40, 1969, 51-65 . RICHTER, H.: Verallgemeinerung eines in der Statistik ben6tigten Satzes der MaBtheorie. Math. Ann.

150, 1963, 8 5 - 9 0 und 440-441. SCHMETTERER, L.: Einffihrung in die Mathematische Statistik, Wien, 1966. SION, M.: On uniformization of sets in topological spaces. Transactions Amer. Math. Soc. 96, 1960,

237 - 246. WALD, A.: Note on the consistency of the maximum likelihood estimates. Annals Math. Stat. 2 0 ,

1949, 595 - 601.

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On the measurability and consistency of minimum contrast estimates