A characterization of contiguous probability measures in the independent identically distributed case

Author(s)

Matsuda, Tadayuki

Citation

Osaka Journal of Mathematics. 13(2) P.335-P.348

Issue Date 1976 Text Version publisher URL

https://doi.org/10.18910/10720

DOI

10.18910/10720

rights

Matsuda, T. Osaka J. Math. 13 (1976), 335-348

A CHARACTERIZATION OF CONTIGUOUS PROBABILITY MEASURES IN THE INDEPENDENT IDENTICALLY DISTRIBUTED CASE TADAYUKI MATSUDA

(Received February 20, 1975)

1. Introduction. The purpose of the present paper is to give a characterization of contiguity of the sequences {Pn>θ} and {Pn,e+en} under a circumstance where for each n PnQ is the distribution of independent identically distributed (iid) random variables (rv's). Contiguity is a concept expressing nearness between the sequences of probability measures. Some characterizations and important consequences of this concept have been established by LeCam (1960), (1966), Roussas (1972) and Philippou and Roussas (1973). Roussas (1972) showed that for stationary Markov process the sequences {Pn,θ} and {Pntθ+Θn} with θn=hn/n1/2y hn->h£:Rfc, are contiguous. Result similar to the above one has been established by Philippou and Roussas (1973) for the independent, but not necessarily identically distributed case. In Section 2, we introduce necessary but not always sufficient conditions for contiguity already obtained by previous authors (see Roussas (1972) and Suzuki (1974)). In the succeeding sections we study the sufficiency of these conditions under several circumstances. In Section 3, we discuss this problem in the case that Pnθ have a constant support under certain regularity conditions. Furthermore, in this case, a simplest condition that \θn\ =0(n~ 1 / 2 ) is shown to be equivalent to contiguity. In Sections 4—7, we deal with the problem of location parameter. After a few preliminary results are established in Section 4, we consider two cases that (1) f(x) = 0 if x

f(x)>0

if x>a

and f(a+0) = 0,

(2) /(*) = 0 if x

f{x)>0

if x>a

and f(a+0) > 0,

in Section 5 and Section 6, respectively, where f(x) stands for a underlying probability density function (pdf). Furthermore, in Section 6, \θn\=o(n~1) is shown to be equivalent to contiguity. Finally, in Section 7, we mention some results which follow immediately from the previous results.

336

T. MATSUDA

2. Necessary conditions for contiguity. For the purpose of completeness of discussions, we present the concept of contiguity introduced by LeCam (1960). Let {(3?, Jίn)} be a sequence of measurable spaces, and let Pn and Qn be probability measures on Jln. {Pn} and {Qn} are said to be contiguous if for any sequence {Tn} of ^-measurable rv's on 3C, Tn-*0 in PΛ-probability if and only if ΓΛ->0 in (^-probability. In order to avoid unnecessary repetitions, all limits are taken as {n}y or subsequences thereof, converges to infinity through the positive integers unless otherwise specified. Also, integrals without limits are understood to be taken over the entire space. If X is a random variable, its probability distribution for a probability measure P is denoted by X(X\P). Furthermore, we write Xn=^X if a sequence of probability measures {Xn} converges weakly to a probability measure X. For each n, let μn be a σ -finite measure dominating Pn and Qn on JHn and write DEFINITION.

(2.1)

fn = dPn\dμny gn = dQn\dμn .

Define the set Bn by (2.2)

Bn = {ωe3f ;fn(co)gn(ω)>0}

and a rv ΛΛ by (2.3)

An = log (£„//„), if = arbitrary,

if

The asymptotic distributions of ΛΛ under both Pn and Qn are independent on the value of Λ* over Bnc. Because we consider only the case that lim Pn (β/)=lim Qn(Bnc)=0 (see Theorem 2.2). Moreover, use the notation (2.4)

J:M = J : ( A , | P W ) .

Theorem 2.1. (LeCam (I960)) If {Xn} converges weakly to a normal 2 2 distribution N(—\σ , σ ), then {Pn} and {Qn} are contiguous. For σ 2 =0, iV(—|σ2, σ2) means the degenerate measure with mass 1 at the origin. Let ρ(Pn, Qn) be the inner product: REMARK.

(2.5)

CONTIGUOUS PROBABILITY MEASURES

337

Theorem 2.2 If {Pn} and {Qn} are contiguous, then (1) and (2) We shall omit the proof of theorem. Because the first assertion of the theorem is just the same as Lemma 5.1, Chapter 1, in Roussas (1972) and the second was proved for the case of independent observations in Suzuki (1974), whose proof extends immediately to the general case. The conditions (1) and (2) of Theorem 2.2 are not necessarily sufficient for contiguity as seen in the following example.X) Let (Ω, Jl, (Jtt), P; (Bt)) be a real Brownian motion with B0=0. Define a stopping time T by τ( ω )=inf {*X0 Bt= — 1}. For any positive integer n, let Pn be the restriction of P to a sub σ-field Jlr/\n defined by Jlτ/Kn={A^:Jl; A Π {τ(ω)^n} e Jln} and let a probability measure Qn be defined as follows dQn=exp {BτAn—^τ/\n}dPn. Then, if i«={ω;τ(ω)\w}, we have Pn(An)->0 while Q»(An)-i±0. Thus {Pn} and {Qn} are not contiguous by Theorem 6.1, Chapter 1, in Roussas (1972). However this example obviously satisfies the conditions (1) and (2) of Theorem 2.2. EXAMPLE.

3. A characterization of contiguous probability measures with constant support. Let Θ be an open neighborhood of the origin of the k~ dimensional Euclidean space JR* and for each 0 e Θ , let pΘ be a probability measure on the Borel real line (R,

AΘ={χζΞR;f(x,θ)>0}.

We call Aθ the support of pθ. (3.2) 1)

Next, for ί e θ we set

φ(x, θ) = This example was orally informed to the author by Mr. Takashi Komatsu. with a slight modification from the paper of Lipcer and Sirjaev (1972).

He got it

338

T. MATSUDA

= o

,

if

and (3.3)

Kn{ω,θ) =

2±ί\ogφ{Xjyθ).

Hereafter, we consider the sequences {Pn} and {£)«} on measurable spaces {(X,

(3-4)

Pn = P Λ , 0 ,

Q« = P",en > where ΘH belongs to Θ for all n. Then the pdf's of Pn and QH are given by

(3-5)

/»H=Π

respectively. write

For simplicity, we write p instead of p0 and P instead of Po and

(3.6)

φn = φ , θn) ,

(3.7)

6φn = \φ{x, θn)dP = \φ(Xjt θn)dP

and (3.8)

An = An(ω,θn).

Lemma 3.1. For any constant any 0^ίan^ll, lim inf (ί-an)n>0

if and only if an = %n~ι).

In particular, lim (1—an)n = 1 if and only if an = oty'1). According to Lemma 3.1 the statements (1) and (2) of Theorem 2.2 are equivalent to the following assumptions (A.I) and (A.2), respectively. Assumption A. (A.I)

1 - ί fix, 0)dμ = oin-1) and 1 - ( JBnl

JBnl

/(*, θn) dμ = o{n''),

where Bnl=A0ΠAθn. 1 (A.2) 1-βφ^Oin- ). Furthermore, we need a set of assumptions for our investigation. Assumption B. (B.I) The set {θn; w=l,2, •••} is bounded and its closure is contained inθ.

CONTIGUOUS PROBABILITY MEASURES

(B.2)

339

For every ίφO,

jl to 0 - Λ M ) I <*#*><>. (B.3)

For every θGθ,

\im[\f{xyθ)-f(xyt)\dμ,

= Q.

(B.4) The set Aθ is independent of 9 G Θ . (B.5) The function

as λ ^ 0 , uniformly on every bounded sets of h^Rfc, where h' denotes the transpose of h. (B.6) T=4β[φ(X)φ(X)/] is positive definite. We use the assumptions (B.I) to (B.3) only to show that 0Λ-»O. If for every 9 G Θ , φ is differentiable in qm (see Philippou and Roussas (1973), assumption (A.2)), then (B.3) holds. (B.4) to (B.6) are the assumptions under which Philippou and Roussas (1973) obtained an asymptotic expansion for the log-likelihood function in the independent, but not necessarily identically distributed case. We will make use of their result restricted to the iid case. Lemma 3.2.

Under Assumptions (A.2) and B, we have | 0 J =0(τΓ 1/2 ),

where the symbol | | stands for the Euclidean norm.

Proof.

By Lemma 1 in Kraft (1955), (A.2) implies

so that |0J-»O by (B.I), (B.2) and (B.3). Since it is enough to consider the case that 0 Λ φO for all n9 we set hn=-^-.

Then we get θn= \θn\hn and \hn\

10*1 = 1. By taking λ to be {|0J} (3.9)

6

and replacing h by {hn}, (B.5) implies

1 -( -l)-hn'φ \θn φn

0,

where φ denotes an abbreviation of

) for all n, it follows from (3.9) and Schwarz inequality that (3 1 0 )
, 2σ 2 ).

By the fact that βφn2=\

1 and (£.6), (3.10) implies

0.

340

T. MATSUDA

(3.H)

^

But (B.6) says that hnThn^δ(>0) desired result.

\ for all n. Thus (A.2) and (3.11) imply the

Theorem 3.1. Let Assumption B be satisfied. Then the following statements are equivalent. (a) The sequences {Pn} and {Qn} are contiguous. (b) liminfp(P Λ ,ρ Λ )>0. (c) \θn\=0(n-^). Proof. According to Theorem 2.2 and Lemma 3.2 it is enough to see that the statement (a) follows from (c). Philippou and Roussas (1973) showed that for θn=7nln1/2y γ

Thus by Theorem 2.1 {Pn} and {Qn} are contiguous. This completes the proof of Theorem 3.1. 4. A characterization of contiguous probability measures with location parameter-preliminaries. Let θ be an open neighborhood of the origin of the real line R, and let f(x) be a pdf with respect to the Lebesgue measure μ on R. Furthermore, we set f(x> θ)=f(x—θ) in order to use the same notations as the previous section. The next two lemmas will be needed in the sequel. Lemma 4.1. Suppose that the following conditions (1) to (4) are satisfied. (1) limn(l-ε

(φn-l)2dp=0.

Then J?(Λ Λ |P)=^iV(-4σ 2 , 8σ2) and consequently {Pn} and {Qn) are contiguous. Proof. (4.1)

For any τ > 0 , we set a«(τ) = n\ J\φH-l\<τ

and

(φn-\)dp

CONTIGUOUS PROBABILITY MEASURES

= n{[

(4.2)

(φn-Vfdp-\\

<

Then (φH-l)dp .

an(r) = n[(φH-l)dp-n\ βφn2^\

But, by the fact that

(4.3)

2

{4

(φn-l)dp} ^p(\φn-l\^τ)\(φn-iγdp -> 0 ,

by (1) and (3).

Therefore (1) gives lim ajj) = -

(4.4)

2 σ

.

Next

But n\(

by (1) and (2).

For every M > τ , we have Jlςpn-

(

= n{

by (3). Therefore (4) gives

so that (4.5)

im n( lim J\φn-l\

{φn-l)2dp

= 2σ2

Next we have (4.6)

KJ\φn-l\<τ

= n[\(φn-\)dp-[ U

«Jl

(φn-

341

342

T.

-^ 0,

MATSUDA

by (1) and (4.3).

By (4.2) and (4.5), (4.6) implies lira σn\r) = 2σ 2 .

(4.7)

From (3), (4.4), (4.7) and Normal Convergence Criterion (see Loeve (1963), page 316), A%

{

By this fact and LeCam's second lemma (see Hajek and Sidak (1967), page 205), we have 2

N{-Aσ\ 8σ ). The desired result then follows. Lemma 4.2. 7/Ίim n(l—gφ»)=0, then we have An -* 0 in P-probability. Consequently {Pn} and {Qn} are contiguous. Proof.

For any 8>0, we have P(IΣ {{φ&i,

θn)-\)-β{φn-\)}

By the assumption, we have Σ {φ{Xj, θn)-\) -> 0 in P-probability. It follows from LeCam's second lemma that ΛΛ -*• 0 in P-probability, as was to be established.

CONTIGUOUS PROBABILITY MEASURES

343

5. Contiguous probability measures with location parametercase 1. In this section, we shall assume the following conditions. Assumption C. (C.I) There exists a real number a such that A0=(a, oo). (C.2) The pdf / is continuous on (a, oo). (C.3) There exist positive numbers d and k such that lim fW k x a * (x—a)

(C.4) lim lim sup sup Theorem 5.1.

=d.2>

f(x+h)~f(x) = 0 .

Under Assumptions A and C, {Pn} and {Qn) are contiguous.

Proof. From (A.2), there exist a subsequence {m} c {n} and σ 2 ^ 0 such that lim m(l—<£φm)=σ2. If σ2—0, then the theorem immediately follows from Lemma 4.2. Thus, assume that σ 2 >0. Since (A.I) implies (2) in Lemma 4.1, it is enough to show that the conditions (3) and (4) in Lemma 4.1 are satisfied. From (A.I) and (C.3), we have (5.1)

θn =

o(n-^+1).

In order to show the validity of (3), we first show that for any given £ > 0 there exists a positive number u such that (5.2)

{x; \φn— l | ^ ε } c ( β , a+u\θn\]\jBmc,

for all sufficiently

large n.

Let g(t, h) be defined by

Then we get

(5.3)

\g(t,k)U±\t\,

k where the constant c depends only on k. that ?7

if

\tU±, L Let η be a positive number such

{x;x>0, \x-l\^e}(z{x; |* 2 -l|^} Π {*; Also, let 7 = 4c—kη

l^-

. From (C.3) there exists a positive number 81 such that

2) This form of assumption was inspired by a lecture by Professor Kei Takeuchi on estimation of location parameter.

344

T. MATSUDA

for all x, a

(x-a)"

.

x

Furthermore, from (C.4) there exist a constant L and a positive number δ 2 such that (5.4)

su sup

f(x+h)-f(x)

<η,

for all h,

\h\<82.

Define the sets Sm, Sn2 and Sn3 by (a,a+281)ΠBnl, m = n2

=

[a+281,L]ΠBnl,

Then we have (5.5)

{x;

}

[φn-l^

i

(jBnlc

u { ( u[( I ψn2-11 ^v) n sni]} u Bnlc Suppose that θn> 0.

If « e Sni, then

(by (5.3))

k d—y

and

Hence, if Λie5 Λ1 and φn2/k-l^-v, we similary obtain x^ ί ? we have

then

^

^0Λ.

In case -δ 1 <(9 M <0,

\θn\ if * e {|^>Λ2//f-11 i ^ } Π 5 m .

Hence, for

CONTIGUOUS PROBABILITY MEASURES

(5.6)

{\φn2/k-l\^v}nSnι(Z(aJa+u\θn\l

345

for all sufficiently large n.

Since f(x) is uniformly continuous on [a+28ιy L] by (C.2) and inf {f(x); [α+2δ 1 ,L]}>0, we have SU

p f\Aχ—θ*)—Ax) II f(x) f(x)

;

XEΞSΔ

)

<η for all sufficiently large n.

Therefore (5.7)

{I ψn— 11 ^v} ΓΊ Sn2 = φ,

for all sufficiently large n.

Furthermore (5.4) implies (5.8)

{I ψn— 11 ^v} Π Sm = φ,

for all sufficiently large n.

Therefore (5.2) follows from (5.5), (5.6), (5.7) and (5.8). By (5.1), (5.2) and (A.I), we get p{{a, a+u \ θn | ])+n p(Bmc)

n p( I φn- \\^S)^n

^ n(d+y)\

(x-aydμ+np(Bnic)

n+l -> 0 , from which (3) follows. We next show the validity of (4). From (5.2), we have lim n[

f(x-θn)dμ = 0 .

This implies (5.9)

lim sup n[

f(x—θn)dμ = 0 .

Since under (3), (5.9) is equivalent to (4), the proof is completed. From Lemma 4.1 and Theorem 5.1, we immediately have the following Corollary 5.1. Suppose that Assumptions (A.I) and C are satisfied. Suppose in addition that lim n(ί—Sφn)=σ2(>0). Then we have

6. Contiguous probability measures with location parameter-case 2. In this section, we consider the other case. We assume the following conditions.

346

T. MATSUDA

Assumption D. (D.I) There exists a real number a such that Aocz[a, oo) and 0 oo). Let/ 7 denote the first derivative of/. (D.3)

(

\f'(x)\dμ

Lemma 6.1. Under Assumptions (A.I) tfnrf D, we have \θn\ =o(n~1). Proof.

Since (a,a+\θn\l

f(x)dμ^ inf {/(*);a

it follows from (A.I) and (D.I) that \θn\ =o(fΓ 1 ). Theorem 6.1. Let Assumption D be satisfied. Then the following statements are equivalent. (a) {Pn} and {Qn} are contiguous. (b) \θn\=o(n->). (c) ΛΛ-*0 in P-probability. Proof. According to Theorem 2.1, Theorem 2.2 and Lemma 6.1 it is enough to see that the statement (c) follows from (b). Let dn be defined by-

Then dn=\\f{x)-f(x~\θn\)\dμ = [

f(x)dμ+ \

^ί

dμ(x)\

Λ

\f'{z)\dμ(z)\

I /(*)-/(*-

\θn\)\dμ

But

= \θn\\

\f'{z)\dμ{z) dμ(x)

\f'{z)\dμ.

It follows from (b), (D.I) and (D.3) that dn=o(n~1). Hence by Lemma 1 in 1 Kraft (1955), 1— Gφn=o(n~ ). The desired conclusion then follows from

CONTIGUOUS PROBABILITY MEASURES

347

Lemma 4.2. REMARK. The argument given above shows that Theorem 6.1 remains valid even if the assumption (D.2) is replaced by the following assumption (D.2') (D.2') The pdf / is continuous on (a, oo) and the derivative/' exists except for finite points on (a, oo).

7. Remarks. In this section, we mention results without proving which follow immediately from Section 5 and Section 6. Assumption E. (E.I) There exist real numbers a and b such that a**
(x— α)*i
= dx and lim -Δ&—k = d2. x /b {b— x) z
Theorem 7.1. Under Assumptions A and E, {Pn} and {Qn} are contiguous. Assumption F. (F.I) There exist real numbers a and b with α<£ such that Aoa[a, b], 00 in P-probability. Acknowledgment. The author wishes to express his hearty thanks to Professor Hirokichi Kudo and Mr. Takeru Suzuki for their valuable comments. WAKAYAMA UNIVERSITY
References [1]
L. LeCam: Locally asymptotically normal families of distributions, Univ. California Publ. Statist. 3 (1960), 37-98.
348
[2]
T. MATSUDA
L. LeCam:
Likelihood functions for large numbers of independent observations,
Research Papers in Statistics (F.N. David, ed.), Wiley, New York, 1966. [3] G.G. Roussas: Contiguity of Probability Measures: Some Applications in Statistics, Cambridge Univ. Press, 1972. [4]
A.N. Philippou and G.G. Roussas: Asymptotic distribution of the likelihood function in the independent not identically distributed case, Ann. Statist. 1 (1973), 454471. [5] C. Kraft: Some conditions for consistency and uniform consistency of statistical
procedures, Univ. California Publ. Statist. 2 (1955), 125-241. [6] M. Loeve: Probability Theory (3rd ed.), Van Nostrand, Princeton, 1963. [7] J. Hajek and Z. Sidak: Theory of Rank Tests, Academic Press, New York, 1967. [8]
T. Suzuki:
The singularity of infinite product measures, Osaka J. Math. 11 (1974),
653-661. [9]
R. S. Lipcer and A.N. Sirjaev: The absolute continuity with respect to Wiener measure of measures that correspond to processes of diffusion type, Izv. Akad. Nauk SSSR
Ser. Mat 36 (1972), 847-889.
**