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.