Cramér-type conditions and quadratic mean differentiability

Ann. Inst. Statist. Math.

29 (1977), Part A, 189-201

CRAM#R-TYPE CONDITIONS AND QUADRATIC

MEAN DIFFERENTIABILITY*

BRUCE LIND AND GEORGE ROUSSAS

(Received Dec. 24, 1973)

Summary

Let (s J) be a measurable space, let e be an open set in R ~, and let {P,; 0 c ~9} be a family of probability measures defined on JZ. Let /~ be a a-finite measure on ~4, and assume that P04</~ for each 0 ~ ~. Let us denote a specified version of dP~/d[~ by f(oJ; 8).

In many large sample problems in statistics, where a study of the log-likelihood is important, it has been convenient to impose conditions on f(~; 8) similar to those used by Cram~r [2] to establish the consist- ency and asymptotic normality of maximum likelihood estimates. These are of a purely analytical nature, involving two or three pointwise derivatives of In f(~;8) with respect to 8. Assumptions of this nature do not have any clear probabilistie or statistical interpretation.

In [I0], LeCam introduced the concept of differentially asymptotically normal (DAN) families of distributions. One of the basic properties of such a family is the form of the asymptotic expansion, in the probability sense, of the log-likelihoods. Roussas [14] and LeCam [II] give conditions under which certain Markov Processes, and sequences of independent identically distributed random variables, respectively, form DAN families of distributions. In both of these papers one of the basic assumptions is the differentiability in quadratic mean of a certain random function. This seems to be a more appealing type of assumption

because of its probabilistic nature. In this paper, we shall prove a theorem involving differentiability

in quadratic mean of random functions. This is done in Section 2. Then, by confining attention to the special case when the random function is that considered by LeCam and Roussas, we will be able to show that the standard conditions of Cram~r type are actually stronger than the conditions of LeCam and Roussas in that they imply the existence of the necessary quadratic mean derivative. The relevant discussion is found in Section 3.

* This research was supported by the National Science Foundation, Grant GP-20036.

189

190 BRUCE LIND AND GEORGE ROUSSAS

1. Introduction

Before s ta t ing the theorems, we m u s t in t roduce the nota t ion to be used. Let (/2, j , P) be a probabil i ty space and denote by

L~(f2, P )=L2( /2 )= {random variables (r.v. 's), X, on (9, ~ , P ) ;

s ~ } .

For X, Y e L~(9), define the inner product (X, Y} as follows

<X, Y > = E ( X Y ) .

Denote by [].]J2 the L~-norm induced by the inner product <., .} ; i.e., for X ~ L~(/2),

[J XEh= (<x, x> ) .

Next , let 0 be a k-dimensional, open subset of R ~ and let g(6)=g(61, 6~,---, 6~), 6 : (61 , 62 , . . . , 6~)' be a random funct ion on (9, j , P) , where the random e lement o~ is omi t ted f rom the nota t ion for the sake of simplicity.

DEFINITION 1.1. The random funct ion g(8) is said to be differen- liable in quadra t ic mean (q.m.) a t 6 when P is employed if the re exists a k-dimensional vector of random functions, g(6), such t h a t

I h 1-11[ g(O+h) -g(O)-h'g(6)]I2---~ 0

as 0 r I h I--* 0, where [. [ denotes the usual Euclidean norm of the vector h ; 9(6) is the q.m. der ivat ive of g(6) at 6. Here, " ' " denotes t ranspose and h'g(~) is the inner product of the indicated vectors.

2. Statements and proofs of so, he theorems on quadratic mean derivatives

The first t heo rem is essentially a special case of a resul t of LeCam [12]. In it, we consider the case where k = l ; i.e., 6 is a real pa ramete r .

THEOREM 2.1. Assume the notation of Section 1. Let g(oJ, O) be a random funct ion defined on ~2 x (9 which is jointly measurable in oJ and 6, and such that ( i ) (~g(o~, 6)/36)I,=,*=g~(w, 6*) ~ L2(O, P"o) for all 6* in some neighborhood

of 6o~8. Suppose furthermore that (ii) g2((o, 6*) is finite, except possibly at countably many points, in a

neighborhood of 8o, a.s. [P"0]"

CRAMI~R-TYPE CONDITIONS AND QUADRATIC MEAN DIFFERENTIABILITY 191

Let h2(O *) = f , g~(~o, O*)Poo(dw), and assume that

(iii) lira t- ' f ~176 [h(u)--h(Oo)ldu=O as (0r t--,O. 80

Then, g2(o), 0o) is the q.m. ([TOo]) derivative of g((o, 0) at 0=00.

PROOF. It follows from the above hypotheses (i) and (ii) and The- orem 264 of Kestelman [8], tha t for every 0,, 02 in a neighborhood of 00 and almost all ([Pso]) o)~/2, we have

g((o, O2)--g(~o, 01)= I ~ (2.1)

Next,

[]t-'[g(~o, Oo+t)-g(o), 00)]ll~

g2(w, u)du .

: f~ t-2[g(w, 80 + t) -- g((o, 00)]2Poo(dco)

(2.2) = i t - ' f 8~ g2(o), u)du 2Poo(dw) j 9 1 JO o

(2.a)

.~- f 80+t t O0+t (2.4) < t -2 h(u)h(v)dudv 00 280

__ f :?

where (2.2) follows from (2.1), (2.3) results from a double application of Fubini 's theorem, (2.4) is obtained by Schwarz inequality and (2.5) by Fubini 's theorem. That is,

I I t-'[g(o), 00 + t) - g(~o, 00)] ll~ =< h(Oo) + t- ' f ~176 [h(u) - h(Oo)]du[ 2, .80

and hence, as ( 0 r we obtain by means of (iii)

lira sup Ht-l[g((o, Oo+t)-g(o~, ~ ~ Oo)] 115-<_ h-(Oo).

On the other hand, by Fatou's lemma, one has

h2(0o)<lim inf IIt-'[g(o,, Oo+t)-g(o,, oo)]11~ as (0~ ) t - ~ 0 .

Combining these last two results, we get

192 B R U C E L I N D A N D G E O R G E R O U S S A S

[It-i[g(~o, 8o+t)-g(o~, 80)] ll~h2(Oo)

Since also

{t-l[g(o~, Oo+t)--g(w, 80)]}2f~ 00)

one has

t-'[g(o~, 8o+t)--g(o~, 8o)]--*g2(w, 8o)

which establishes the desired result .

as (O:/:)t--~O.

as ( O r

in q.m. ([P~o]) as ( O r

All of the hypotheses of the theorem can be readily checked except, perhaps, for (iii). The following lemma will be helpful in fo rmula t ing an a l te rna t ive condition.

LEMMA 2.1. Let q: R---~R, and let xo be a cont inui ty point of q. Let [ q ] be Lebesgue integrable in a neighborhood of xo. Then, as (0 g=) t--* O ,

t ~~ ]q(u)--q(Xo)[du=O. lira t-i J x 0

PROOF. Since q is continuous at x0, for a given ~>0, the re exists a a>O such t h a t

1q(u)--q(xo)l<~ if [u--xo]<~.

I f we choose [t [<a, we have

t -1 f ~~ [q(u)--q(Xo)]du<z J x o

and the l emma is proven.

Using the above lemma we see t h a t (fii) is implied by (iii)' h(8) is locally integrable and continuous at 80.

To close this section we state, wi thou t proof, a t heo rem which has been der ived in an earlier paper (Lind and Roussas [13]). I t deals wi th the case k > l , and will be re fe r red to in Section 3.

THEOREM 2.2. For each 8 ~ O, we assume that the partial derivatives in q.m. ([P~]) of the random func t ion g(8) exist and are continuous (in 8) in L~-norm, H" 112. Then {7(8), the derivative in q.m. of g(O), exists and is equal to the vector of partial derivatives i n q.m. That is, f o r each 8 ~ 0 and hr

[hi -1 g (O+h) - -g (O) -Z h~oj(O) ---*0 j = l 2

as IhI---~O, where gj(8) denotes the partial q.m. derivative of g(8) at t~.


This theorem changes the problem from a search for a vector of functions and the taking of limits as a vector variable approaches zero in norm to one of finding the partial derivatives in q.m., which involves only a single function and a real a rgument , and then checking to see tha t they are continuous in [].l[2-norm. Theorem 2.1 may serve as a way of finding the partial q.m. derivatives.

F u r t h e r use of these theorems will be demonst ra ted in a forthcom- ing paper which will be devoted to the calculation of some asymptotically optimal t e s t s for certain failure distributions.

3. Comparison of Cram&r type conditions with the conditions of LeCam and Roussas

In LeCam [12], a set of conditions of the Cram~r type is shown to imply a certain differentiability in quadratic mean assumption of LeCam and Roussas. This set of conditions is essentially minimal and the proof is quite involved and makes use of some techniques of functional analysis. In this section, it will be shown that, if one instead chooses the stronger conditions usually made, then the conclusion can be arrived at by some

simple arguments. To facilitate the comparison of the two types of conditions for the

remainder of this section, let (Y2, J ) be (R, _~) the Borel real line, let 0 be an open subset of R ', and let {P,; 0~0} be a family of probability measures defined on _q~. Let /2 be a a-finite measure on _~, and assume tha t P,<<ff for each 0 ~ O. Let u s d e n o t e a specified version of dPo/d,~ by f(x; 0). Let 2(1, X2, . . . be a sequence of independent, identically distr ibuted random variables defined on (R, ~ ) . Then the assumptions of Roussas [14] are the following: (A1) The set on which f ( . ; 0 ) is positive is independent of 0.

Set r O*)=[f(X,; O*)/f(X,; 0)] '/2. Then (A2) ( i ) For each 0 e O, the random function r 0*) is differentiable

in q.m. with respect to 0* at (0, 0) when Po is employed.

Let r be the q.m. derivat ive of r 0*) with respect to 0* at (0, 0). Then

(ii) ~{1(0) is Xl- '(_~)• where 2 is the Borel a-field in R' and C is the a-field of Borel subsets of O.

(iii) For every 0 e O, F(O)=4E,[r162 is positive definite. We now state a set of conditions of Cram~r type. These are found in Davidson and Lever [3]. (B1) Same as (A1) above. (B2) For almost all ([/2])x ~ R and for all 0 ~ O,

0 in f/aO~, 82 In f/ao~ao~ and ~3 In f/00~30~00~


(B3)

(]34)

(B5)

exist for r, s, t = l , . - . , k. For almost all ([ff])x E R and for every O E O,

18fl~O~ [< F~(x) and I a f f lao~s I< F~(x) ,

where Fr(x) and F,(x) are integrable over R, r, s = l , - . . , k. For every 0E O, the matrix I(o)=(LW~)) with

a l n f a l n f s, x o ) = c , ( - - - /

is positive definite with finite determinant . For almost all ([ff])x E R and for all 0 c O,

a a l n f o <H~(x) a<ae, ae~

where there exists a positive real number M such tha t

Eo[H~.~(X~)] < M < oo

for all 0 E 8 and r , s , t = l , . . - , k . These assumptions are simply a mult i -parameter given by Cram~r [2].

extension of those

When one wants to derive more delicate results for a family of distributions it is convenient to add some fu r the r hypotheses on the density. One purpose of these fur ther restrictions is to insure the con- t inui ty (componentwise) of I(0). (See Kaufman [7], Weiss and Wolfowitz [20] and LeCam [9] for papers referr ing to estimation, and Wald [18], [19], Davidson and Lever [3] and LeCam [9] for papers referr ing to tests of hypotheses.) Since the conditions (A), a long with the assumption tha t F(0) is continuous, are sufficient to provide results of the same nature, it is not unreasonable to incorporate a condition which implies the continuity of I(0) into (B) above. The one we choose to use is from Davidson and Lever [3] (see also Wald [18], [19]). (B6) There exist positive real numbers ,~ and T such tha t whenever

IIO"-o'11~ ~, 107-<' I<,~, 0', 0- e o , r = l

&-> [ (a~ ln , f l h2- i ~'L\~lo,,) j < T < o o for r, s--l,..., k.

Since the proof tha t (]36) does, in fact, imply tha t I(a) is continuous is quite interesting, and not included in the Davidson and Lever paper, we include it here. In other words, we establish the following result

THSOREM 3.1. Under assumptions (B1)-(B6)


Oln f O l n f I(0)= ~ ' o - - -

is continuous in the sense that each component is a continuous function of O.

PROOF. It is well known that (B1)-(B5) imply that

I(0)=( P / a 2 1 n f [

Therefore,

- O lnf( ; o) of( ; O lnf oof( Oo)dZ( ) 30~00~ 00,30~ '

= fR [02 In f oof(X; 0o) O~ In f of(x. O)]dlx(x).

By Taylor's theorem, we have for almost all ([~])x ~ R,

o - ~ k a 8 In f o* O~lnf O21nf § (0~--0o~) 30rO0 s 30rO0 s t = l OOrOO~OOt '

where O* lines on the line segment joining 0 and 0o. From this, have

a ~ l n f of(x; O)-- a~ l n f Oof(X; 0)-t-~ (O~--Oo~) a3 l n f o.f(x; O) O0,cqO~ ao,ao~ t =i aO,~OflO~

which implies

w e

a21nfoof(X;Oo) a21nf ef(x;O) 30,30~ ao,ao~

80 k O* --321nf [f(X;Oo)--f(x;O)]--Z(O~-Oo~) 331nf f ( x ; O ) .

~Or~O s t = l 30r~Os~O ~

Thus,

I L,~(o)-L.~(Oo) I

i 021nf Oo O)]dt2(x) _< If(x; Oo)--f(x"

+ ~(Ot--Oot)fR O~lnf o. f(x;O)d/x(x) t=t 3O,aS flO~

< i [ a~ ln----L Oo I I f ( x ; Oo)--f(x; O)1'121f(x; Oo)--f(x; O)ll/~d,~(x)[ = JR I ~Or~O~

+~']0~--O0~]fR[ aa ln f o* f(x;O)dt2(x). ~= I 38 ,30 fl0 ~


By the Cauchy-Schwarz inequality this is less than or equal to

{f 021nf I~[f(x'O~ ),,,2i . . 08,08~ .o d t4x)

" {fR i f(x; 8o)--f(x; O)[d,a(x)} '/2

+~]0~--00~liR[ 031nf o. f(x'O)dix(x). ~=l 08~08~08~ '

By assumption (B5), the last sum above is bounded by M[]O-8ol[. The first t e rm is bounded by

C I f ~. 1 ~ oo f (x ; 8o)dtz(x)+ . ~ . o f ( x ; 8)dv(x)

a f ,2 �9 jo,-Oo j <',*')}

where 8** lies on the line segment joining 8 and 00. By (B6), the first factor above is less than (2T) In for 118-801[ sufficiently small (<~). By

the second factor is less than (KIIS-8oN) ~/~, where K = f , F~(x)dt~(x). (B3),

Therefore I/r,~(0)--Ir,~(80)l<=MliS--80[]+(2Tki[8--80il) ~12, provided 118-801[ < , . From this it follows tha t lim]L,,(8)-L,~(8o)[=O, as 0--*00, as was to be seen.

Let us now assume tha t k - l , and prove the following theorem.

THEOREM 3.2. In the notation above, assumptions (B1)-(B6) are stronger than assumptions (A1)-(A2) in the sense that any density which satisfies (B1)-(B6) also satisfies (A1)-(A2).

Before proceeding with the proof of Theorem 3.2, we specialize Theorem 2.1 to the case where the random function is r (In the notation below (i), (ii), and (iii)' correspond to (i), (ii), and (iii)' as used in Theorem 2.1 and the discussion following it.)

Recall tha t

Therefore,

r ; 80, 8*)= [f(x ; 8o)]-l/2[f(x ; 8*)] ''2 .

3r 0o, 0") o*=~-- 1 Of(x; 8*) ~[f(x; 8o)f(x; 0)] - 'n �9 00* 2 aS*

Thus, we want to have

1 ~" [(0f(x ; 8)/08)[,.]~ ( i ) --s ). R f ( x ; 8')f(x;Oo) f (x; Oo)d/~(x)

CRAMER-TYPE CONDITIONS AND QUADRATIC MEAN DIFFERENTIABILITY 197

(ii)

4 30 ' '

1 I(O') 4

finite for 0' in a neighborhood of 00;

For x (almost all [/~]) we wan t

1 [f(x" 00)] -in Of(x ; O) o / [ f ( x ' 90 ',

; 0')] 1/2

to be finite as a funct ion of 0' (except for possibly countably many values) in a neighborhood of 00;

(iii)' I(O) is a continuous funct ion of 0 in a neighborhood of 00 (since this in addition to (i) above implies the local integrabi l i ty of h(O) = (1/2) [I(0)]m).

In proving Theorem 3.2 let us note t h a t verification of (i), (ii), (iii)' above for each 00 ~ 0 is sufficient to establish the following

LEMMA 3.1. Conditions (B1)-(B6) imply that (A2)-(i) is satisfied, with

1 3 In f ( x ; 0') 0,=, Cdx; 0)= r 0")[o*=o-- 2 30*

for each 0 ~ O.

PROOF. Clearly, (i) is satisfied because of (B4). Let us consider condition (ii). Now,

aCdx; 00, 0 " ) 0 . = o , = l [ f ( x ; 0o)]-i/2[ 3 f ( x ; O*) o.=~,l[f(x" O')]-In . 30* 80" '

By (B2) (9 ln f)/90 exists and it follows t h a t both [(31nf(x; 0)/90)],,] and [f(x;O')] 1/~" are finite funct ions of 0' and hence (ii) is satisfied. For (iii)' it suffices to show t h a t I(O) is a continuous funct ion of 0. This has been done in Theorem 3.1.

PROOF OF THEOREM 3.2. To complete the proof of Theorem 3.2 we observe t ha t (B1) and (A1) are identical. I t only remains to show tha t (A2)-(ii) and (A2)-(iii) are implied by conditions (B). F rom L e m m a

3.1 we have r = (1/2) ((3 ln f ( x ; 0))/90), and thus (A2)-(ii) and (A2)-(iii) follow f rom (B2)-(B4). This completes the proof of Theorem 3.2.

We thus know tha t for k = l , conditions (B) imply conditions (A). The above result also shows tha t the part ial derivat ives in quadrat ic mean are given by the pointwise part ial derivat ives under conditions (B). I f we could also show t h a t conditions (B) imply the cont inui ty in


L2-norm of these partial der ivat ives we would have the existence of 4, in the ease k > l upon application of Theorem 2.2.

W h a t we mean by cont inui ty in L2-norm in this ease is the following: Since we have already shown tha t

l ( Of(x; O) ) / [ f (x ; Oo)]

where r d00) denotes the part ial q.m. der ivat ive of r 0o, 0") wi th respect to 0* at 0"=0o, wha t we wan t to show is t ha t

lim i fR I( 0 f ( x ; 0) ) I f (x" O*)- (Of(x; O)Ool//f(x. Oo)l 2 o.4Oo 4 00~ o. ' 00~ , '

�9 f (x ; Oo)dtt(x)

equals zero. This is equivalent to

l i m l I31n f(x;O) o. Oln f(x;O) OoJ2f(x;Oo)dl2(x)=O . o.-o o j R 00~ 00~

Since (32 In f)/OOrO0, exists s = l , 2 , . . . , k, we have

a l n f o* a l n f o0 0Or 00r

Thus if we can show

- - * 0 as 0 " - - * 00 �9

a In f 2 . ]

the proof will be complete by Vitali 's theorem. (B6) is not quite s t rong enough to readily give us the desired result . W h a t is needed is a re- qu i r emen t such as is found in Bahadur [1], Roussas [15], or Schmet- t e r e r [17]. (B6)' For any given 00 in O, and r = l , . . . , k, the re exists a neighbor-

hood, No o, of 00 and a _~-measurable funct ion Mdx) such t ha t

O l n f <=Mr(x) for all x ~ R 3Or

and all 8 ~ No 0 and such t h a t

Eo0M ~<oo , r = l , . - . , k .

Wi th this condition it is obvious f rom the domina ted convergence theorem t h a t

f Valn:l L aO, Io, d ;6~

CRAMt~R-TYPE CONDITIONS AND QUADRATIC MEAN DIFFERENTIABILITY 199

is a continuous function of 0* in a neighborhood of 00.

4. Closing remarks and an example

The results presented in this paper jus t i fy earlier assertions tha t the assumptions (A) are actually weaker than those commonly occur- r ing in the l i tera ture and not just of a different nature. This shows tha t the results of Roussas [14], [16] and Johnson and Roussas [4], [5], [6] extend earlier results in two ways. The regular i ty conditions are weaker and the assumption of independence is also dropped. Since there are examples in which conditions (A) are satisfied when conditions (B) are not, the results actually have wider application as well.

Example. As an example of a density which is, clearly, non-regular wi th respect to the usual Cram6r type conditions, but is regular with respect to the (A) conditions, consider

(4.1) f(x; 0 ) - 1 exp [ - l x - 0 ] ~} , 1/2<2_<_1. 2/ '(1+1/2)

A discussion of the case 2=1, the Laplace distribution, is found in Johnson and Roussas [4]. We include here a brief discussion for 1/2< 2<1.

For the density (4.1), we have

-~1 XI--O*I'+I[xI--OI~} r 0") ---- exp ,-- [ [

Now let

gl(0)=-

- ~ , ~ ( o - y o x~<o

0 XI=O

1,~( x , -o) x~ > o .

I t is clear tha t

h-1[r O+h)- l]--~gl(0) in P~-probability

as (0:#) h--~0. Next,

o~162 O+h)-l]}2=2h-2[1-Eor O+h)] .

Thus, if we can show tha t

(4.2) 2h-211-E0r 0+h)]--*g'0[gl(8)] 2 as (0 r h--*0,


we will have verified (A2)-(i), and the remaining (A) conditions are

obviously satisfied for r A straightforward computation gives

E0[g,(0)]2_ gF(2-- 1/~) 4r(1 + 1/0

To verify (4.2), we first form the expression 2h-'~[1-g'o,41(0, O+h)]. This is equal to (after some simplification)

2h-211 1 f~ [--~(y+hy-ly~]dy F( l+ l /2 ) .o exp

Evaluation of the limit, as (0~:)h approaches zero, can be accomplished by two successive applications of L'Hospital's rule (which is easily justi- fiable by the Dominated convergence theorem, differentiation of the second integral in the expression immediately above following by Lei- bowitz's rule). The result of these operations yields (;~F(2--1/,O)/(4F(l+ 1/~)) and the example is completed.

UNIVERSITY OF WISCONSIN

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CRAMt~R-TYPE CONDITIONS AND QUADRATIC MEAN DIFFERENTIABILITY 201

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Cramér-type conditions and quadratic mean differentiability

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