SUZUKI, TAKERU

Osaka City University Repository

Suzuki, T.
Osaka J. Math.
25 (1988), 115-122
ON A RELATION BETWEEN HIGHER ORDER
ASYMPTOTIC RISK SUFFICIENCY AND HIGHER ORDER
ASYMPTOTIC SUFFICIENCY IN A LOCAL SENSE
TAKERU SUZUKI
(Received November 26, 1986)
(Revised May 7, 1987)
1. Introduction. In Takeuchi [4] higher order asymptotic risk suffi-
ciency of maximum likelihood estimator has been discussed. In this paper
we try to find some relations between asymptotic risk sufficiency with a special
loss function and asymptotic sufficiency in a local sense.
Let 5>
Λ
= {PQt n θ e Θ} be a family of probability distributions on a measurable
space (3?, cJ?
Λ
) with an index set Θ which is a subset of an Euclidean space with
the usual norm | | . For a sub σ -field C of Jl
n
, real number c^O and θ, 0'eΘ
let tζ(c: θ, <9'Wnf (l+cΓlil—EP (φ)+cEP , (φ); φ are ^-measurable statis-
θ,n 0 ,n
tical test functions on 3?}. We note that rζ(c: θ, θ') means the Bayes risk of
statistical problem of testing a hypothesis 'P
β
/fll is true' against an alternative 'Pβ,n
is true' with experiment (3?, C, |FV,«> P0,«}) relative to a prior probability distri-
bution (cl(l+c), l/(l+c)) on {#', θ} provided that the loss function is simple.
Let {&
n
\ if=l, 2, — } be a sequence of sub σ-fields of iJl
n
}(<B
n
c:Jl
n
). In
this paper we give a sufficient condition about the Bayes risk r^n for {&„}• to be
higher order locally asymptotically sufficient sequence of σ-fields. More
precisely our main result in this paper is the following: Under some condi-
tions if for some positive number a sup sup sup {r$n(c: θy θ*)—00 Θ*<=K θ .-H1/2 |0-0*|^δ
r^»(c: θ, θ*)} =o(n~*) for every b>0 and every compact subset K of θ, then for
every β satisfying 0<y8<3~1α{^
Λ
} is locally asymptotically sufficient for {ί?*}
with order o(n~β) in the sense that for each n=l, 2, ••• and each 00eθ there
exists a family {Ql*
n
\ 0eθ} of probability distributions on (3?, cjϊ
β
) for which
3>
n
 is sufficient σ -fieJd and that for every
uniformly in 00 over every compact subsets of θ. Here IHIj^ means the total
variation norm over Jl
n
.
We have discussed such a problem in the case a=β—Q in Suzuki [3] under
116 T. SUZUKI
non-local situation. In LeCam [1], Chap. 5 he discusses some relations between
insufficiency and deficiency in his terminology.
In Section 2 some auxiliary results about the order of asymptotic suffi-
ciency are proved. The main theorem is stated and followed by some discus-
sions about the asymptotic sufficiency in non-local sense in Section 3.
2. Auxiliary results. For each n<=N= {1, 2, •••} let S
n
= {P
θ
,
n
; 0<Ξθ}
be a family of probability distributions on a measurable space (3£, <Λ^) with an
index set θ. For a subset t/(Φφ) of θ we shall denote by S^ the totality of
-P0f,/s satisfying Θ^U. We assume that for each n^N 3
>
n
 is dominated by a σ-
finite measure μ
n
 on (2£y <_Λ?Λ). The probability density function of Pθ>n relative
to μ
n
 will be denoted by p
n
(x, θ). Without loss of generality we assume in the
following that μ
n
 is a probability measure on (3?, <JL
n
). For each 0, 0'eθ let
Sn(θ)= {*; p
u
(x, 0)>0} and let h
n
(x; θ,θ')=ρ
u
(x, θ)lp
n
(x, θ') if
For each θ, 0'eθ and real number s^>l we define
We note that β
n
(θ, θ') = 1-/M(1 : θ, θ').
Let {U
n
\ be a sequence of nonempty subsets of θ. For {U
n
} we consider
the following assumption.
ASSUMPTION 1. There exist a sequence {0*}
 n
^N(θϊ e Un) and a positive
number γ such that
(a) Foi every s^l
Km sup sup J
n
(s: θy
(b) sup β
u
(θ9 θt) = o(n~
For a sub σ -field C of Jl
n
 we denote by Φ(C) the family of ^-measurable
statistical test functions on 3C. For each θ,θ'^& and each real number ^^0
we define
rζ(c: θ, θ') = inf (l+c)-i{l-EPtιf (φ)+c£,.,t. (φ); φeΦ(C)} .
Let {^
a
} be a sequence of sub σ-fields of {<Λ
n
} (<B
n
d o?»). For each
define jf,(«, ^)=^μB|j>B(^, ^)|-®»] the conditional expectation of pn(x, θ) given
with respect to
 μ* and put 5ί(^)={Λr;^,,(Λ?, (9)>0>. For θ,θ'eθ define
(*; θ, θ')=f
n
(x, θ)lf
n
(x, θ') ifxeS'
n
(θ'), = + oo if xeSWYnSίW, =1 if
Sί(ΘJΓ\SίKΘ). For c>0 and δ>0 let EH(c, θ, S)= {*;£,(*; β,
and £ί(c, β, S)= {*;&,(*;
HIGHER ORDER ASYMPTOTIC RISK SUFFICIENCY 117
Proposition. Suppose that for some positive number a, and a sequence {#*}
 nejv
(2.1) sup sup ir* (c: θ, θ*)-r^(c: θ, θ*)} = o(n~*) .
C>0
Then we have
(2.2) sup δ(l+c)-1 \
u
(c, δ) = o(O » and
c>0,<5>0
sup δ(l+c)"1 \n(c, δ) — o(ri~*)
where \
Λ
(c, 8)= sup Pf
n
.
n
(E
u
(c, θ, δ)) and \'
Λ
(c, 8)= sup P9lu(E'u(c, θ, δ)).
This proposition can be proved in the same way as the proof of the first
and second steps of Theorem 1 in Suzuki [3]. So we shall omit the proof of
the proposition.
Theorem 1. Suppose that Assumption 1 is satisfied with a sequence {#?}„<=#
and γ>0, and that {<B
n
} has the property (2.1) with β>0. Then for every β
satisfying 0<β<3~1a and β^j, {<B
n
} is asymptotically sufficient for {9?nn} with
order o(n~β) in the following sense: For each n^Nthere exists a family {q
n
(x\ θ, θ%)
of probability density functions on (3£, Jl^) relative to μ
n
 such that
(i) each q
n
 can be factorized as follows:
n (v ft fl*\ — r (v fi fl*\ /> (v fl*\q
n
\X, V, U
n
 )  rn\x> V) "n ) Pn\x> σu )
where r
n
 is a ^^measurable function, and
ΊΛ(M)-ί.(*;MΪ) I <*/*.=
Proof. We shall divide the proof into several steps.
The first step. Suppose that Assumption 1 is satisfied with a sequence
Whs* and γ>0, and that {$
n
}
n(ΞN has the property (2.1) with α>0. Let β
be any number satisfying 0<β<3~~1a and /3^γ. Take £
x
 be any number
satisfying 0<51<3~
1
 (α—3/3). Let a
n
=n~
β(log n)"1, m
n
—n*ι and iH=[mnaήl]
+ 1 where [a] means the maximum integer not exceeding a. Put (%,=) 7
n
(xi (9, θ*)= \h
u
(x; θ, θ*)-g
n
(x: θ, θ*) \ , (rί=) Ύ'n(*', θ, fl*)= \h
n
(x; θ, θ*)-IWu(x)
g
n
(x ,θ,θ*)\ and
where W
n
=W
n
(θ, θ$)= {x\g
n
(x\ θ, θ$)^m
n
} and IWn means the indicator func-
tion of W
n
.
We have
118 T. SUZUKI
sup
 Pn(θ, W)^su£ J^ <yndPfa+tup \wM*'> θ> W <&•:.* =JΪ+JΪ* >
(2.3) and J* = sup j γM dPθ*ιn^an+ sup j ^ <γndPθ*ntn = c
(D
n
= {x;y
n
^a
n
}).
Furthermore we have
— sup I fγ
n
dP
θ
*
n
^ sup \ Ύ
Λ
^β*
Λ
+ sup I 7
- -\JD
n
ι\w
n
 "' θ<=U-4DnnwH "' θ<=U~lwH
where W'.= {x\ h
n
(χ-, θ, θ*)^m
n
}, W
n
=W
n
D W'
n
 and W*=W
n
Γ\(W'
n
γ.
The second step. It holds that
Γ
 2/n
"
2
 Γ
Iί= sup ^ r.dP :.. ^  Σ sup ΎndPeln
θςΞU
n
*
D
nW
n
 i = l θ<=U
n
*Bi
(2.4) +13 sup ( γ
n
rfP
β
;,n
' - o ^ e ϋ J c ,
where 5y=fFnn {^  An(^; β, 0*)^2~\i+l) an,tu(x; θ, θί)<2'li an} and C~
W
n
n {^h
n
(xιθyθϊ)<2'\i+l)anίgn(x'yθ9θ^^2^(i+2)an}. Using the pro-
perty (2.2) in Proposition we can evaluate /ί§l (ι'=l, 2) as follows. Taking ac-
count of 3f!<α— 3/3 we have
/ί.ι= ίi/e^L7"^-11
£2i
n
m
n
[ sup sup P*;,
Λ
{*; A
Λ
(Λ:; 5, 5*)^2"1(/+1) a
n
(2.5) ^2ί.»,[ sup
l^*^2,
 n
-
rt[ sup
^^ n"
(β>
"
2p
"
3t
ι
)(log n)2 η'
n
 (Al is a constant)
Similarly we have
(2.6) /ί.2 =
Thus from (2.4) and (2.5) we have
(2.7) I'
m
 = o
HIGHER ORDER ASYMPTOTIC RISK SUFFICIENCY 119
The third step. Next we evaluate I'
n
 as follows. For every $>1 we have
/ ί / =su f .y.rfiP^sup \ Λ,(*;
*
W
n
 q
 Θ^U
n
^h
n
>m
n
}
Hence we have
I'
n
'^A2(s)(mnΓ* = A2(ήn«-*ι
where A2(s) is some constant depending only on s. We can choose s>ί large
enough so that
(2.8) /£' = o(n-') .
From (2.7) and (2.8) we have
Hence from (2.3) we have
/* = o(n-').
Put W'
n
'= {x'9 hn(x\ θ, θ^)<2~1 mn} . Then we have
7**=βup ( Λ(*;M«*W;.«
ΘeU
n
Jw»
(2.9) ^sup [
 c //A.(Λ?;β,β*)dP f.. f l+8up f c H
θ^ϋn
ίw nw
 θ<=U
n
lw
n
n(w
n
)
^Z-1 m
n
 x;(ifi
e> mnl2)+(mn/2γ-s sup /β(ί : θ, θ*)
The first term on the right hand side is of order o(n~β) by Proposition. The
similar consideration as the evaluation of I" implies that the second term of
(2.9) is also of order o(n~β) for sufficiently large number s. Thus we have
Hence it follows from (2.3) that
(2.10) sup p
n
(θ, θ*) = o(n-*) .
The fourth step. Let atf, ^*)=[^ /r.(*)ί.(*; ^> W) ^ C-Π^ °°) and
letr.(*;β,βί)=α.(d, W) /*„(*)*,(*; «, W)if β.(β, β?)<oo ,=1 otherwise. De-
fine ?B(Λ;; 5, 0*) =rs(ic; ,^ θf)pn(x, θ%) and let £>g»B be the probability distribution
on (3C, tj?
Λ
) with density q
n
(x; θ, θ$) relative to μ
n
. We note that &„ is sufH-
120 T. SUZUKI
cient σ-field for the family {Q$
n
 θ e θ} by the factorization theorem. It follows
from (2.10) thai there exists nQ such that an(θ, 0*)<oo for every n^fjQ and every
Θ^U
n
. Therefore we can assume without loss of generality that a
n
(θ, 0*)<oo
for every θ&U
n
 and every n^. 1.
Under this circumstances we have
\PA*, *)-?.(*; θ, θ*)\dμ
Λ
(«*> ' *'(*; θ' ^-".(O, *?) Iw
u
(*)gn(*\ θ, tit) I
p.(x,θ*)dμ.+βJ(θ,θ*)
= P«(θ, θ*)+ 1 1 -a
n
(θ, θ*)-1 1 +β.(θ, θ*)
Here \\v\\ji means the total variation norm of a signed measur v on (2C, <Jl
x
).
From Assumption 1, (b) and (2.10) we have
Th's completes the proof of the theorem.
3. The order of local asymptotic sufficiency. In th's section the
index set θ is assumed to be a subset of />-dimensional Euclidean space Rp. We
denote by | | the usual Euclidean norm in JR°. For 0eθ and δ>0 let U
n
(θ, b)
Let {^B
n
}
neN be the sequence of sub σ-fields <Bnc:Jln as in the previous
section. We consider the following assumption which will be used to prove
our main theorem, Theorem 2.
ASSUMPTION 2. For every compact subset K of θ and b>0
(a) lim sup sup sup JH(s: θ, 0*)<oo (V s>l) , and
(b) sup sup
Θ*(=K
Let a be a given positive number. We state a result about higher order
locally asymptotic sufficiency of {*B
n
} for
Theorem 2. Suppose that Assumption 2 is satisfied zΰith γ>0, and that
for every compact subset K of θ and every δ>0
(3.1) sup sup sup {r®«(c\ θ, θ*)—r<Λn(
c:
 θ, (9*)} = o(n~*) .
00
Then for every positive number β satisfying /3<3-1α and β^Ύ {*B
n
}
neN is locally
asymptotically sufficient for {&
n
} with order o(n~β) in the following sense: For each
HIGHER ORDER ASYMPTOTIC RISK SUFFICIENCY 121
n^N and each ΘQ^& there exists a family Qn°— {QfynΊ θ&θ} of probability distri-
butions on (3£} <Jln) such that
(i) ,®
n
 is sufficient for Qfy , and
(ii) for every compact subset K of Θ and every ό>0
sup sup ||P
β
.n-£*MU =o(n~t).
°*
Since the above result follows directly from Theorem 1 we shall omit the
proof.
It is open problem whether non-local version of Theorem 2 still holds or
not, i.e., whether any conditions such as in Theorem 2 imply the followings
or not: There exists a sequence Q
n
={Qe,n'y θ^®} of probability distributions
on (3£y <Jln) such that &n is sufficient for Qn> and that for every compact subset
Kof®
(3.2)
The case of a— β=0 has been discussed in Suzuki [3] in such a non-local
situation.
It is well known that under some regularity conditions there exist a sequence
{v
n
}
nfΞN of estimators of Θ, a positive number γ and a number v^l having
the following property: For every compact subset K of Θ there corresponds
a(K) such that
sup P..Λβ"*|,(*)-0| ^a(K) (log «)°/2} = „(»-»)
Θ&K
(c.f. Matsuda [2], Chap. 3).
Using such an estimator {#„} we may be able to construct {Qe,
n
> θ^&} satisfy-
ing the property (3.2), and for which <B
n
 is sufficient.
Acknowledgement. The author wishes to express his hearty thanks to
the refree for many helpful comments. The evaluations of (2.3) and (2.9) have
been much simplified following the suggestion of the referee.
References
[1] L. LeCam: Asymptotic methods in statistical decision theory. Springer-Verlag,
New York, 1986.
[2] T. Matsuda: Tokeiteki suisoku no zenkinriron (Asymptotic theory of statistical
inference). (In Japanese), Hakuto Shobo, Tokyo, 1985.
[3] T. Suzuki: A characterization of approximate sufficiency. Memoirs of the School
122 T. SUZUKI
of Science & Engineering Waseda Univ. 45 (1981), 113-122.
[4] K. Takeuchi: Higher order asymptotic efficiency of estimators in decision procedures.
Statistical Decision Theory and Related Topics III. S.S. Gupta and J.O. Berger,
eds. Academic, New York, 1982.
Department of Mathematics
School of Science and Engineering
Waseda University
3-4-1 Ohkubo, Shinjuku-ku
Tokyo 160, Japan


On a relation between higher order asymptotic risk sufficiency and higher order asymptotic sufficiency in a local sense

Suzuki, Takeru

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On a relation between higher order asymptotic risk sufficiency and higher order asymptotic sufficiency in a local sense

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