数理解析研究所講究録
第 538 巻 1984 年 47-64$\dot{\sim t}_{j}\llcorner$

台が有限な分布の位置母数の同時推定

慶応大工 赤井 豊秋(Toyoaki Akai)

Abstract

Suppose that components of a rand $om$ vector $X=$$(X_{1}$....

,$X_{p})’(p\geq 3)$

areindependently distributed, each having a probability density which is positive ona finite interval, is symmetric aboutof the location vector$\theta=$$\theta_{i}$and has the same variance. In estimation$(\theta_{1}, \ldots , \rho_{p})’$under the squared loss function explicit

estimators which dominate $X$ are obtained by using integration$b\gamma$

parts toevaluate the risk function. Further, explicit dominating estimators are given
when the distributions of cornponents are mixture of two uniform distributions.

For the loss function$L(\hat{\theta}.\tau 1)=|$}$\hat{\theta}arrow\theta||^{4}$such an estimator is also given when the

distributions of components of $X$ are uniforrn distributions.

1 Introduction.
Stein (1955) and Brown (1966) proved that the best invariant estimator of
the location vector of three or more dimensions are inadmissible, and there has
been considerable interest in how to improve it. James and Stein (1961).

presented an explicit estimator $\{1 -(y-2)/||X\Vert^{2}\}X$ vhich is better than $X$

under squared error loss if $X$ has a normal distribution $wi$ th covariance matrix I,
the identity rnatrix. They also showed that the estimatorノ$A_{\}}\wedge^{\cup}$(1. 1)$\delta_{1}(X)=\{1-$is better than X. without the normality assumption, for sufficiently small
sufficiently large

$a$

$b$

and

. They did not, however, determine explicitly the values ofthese constants.

When $X=(X_{1}, \ldots.X_{p})^{l}$ is an observed value from a spherically symmetricp-dimensional distribution, explicit estinators of a location vector which dominate $X$ are given. See Brandwein and Strawderman (1980) and the papers intheir references. Shinozaki (1984) obtained similar results in the case wherecomponents of$X$

are independent, identically and symmetrically distributed$p$randorn variables, by applying integration by parts to three typical distributions; umiform, double exponential and$t$. Since Stein (1973) used integration byparts for estimating the location parameter of the normal distribution, it has
been shown to apply to simultaneous estimation problems in general continuous
exponential family by many authors. See Hudson (1978).

In this paper, we assume that components of$X=(X_{1}\ldots..X_{p})$ ‘areindependently distributed, each having a probability density which is positive ona finite interval, is symmetric about$\theta_{l’}$. the center of the interval, and has thesame variance, and we estimate the location vector
estimator$\delta_{1}$$\theta=(\theta_{1}, \ldots, \theta_{p})’$by theof (1.1) under the squared error loss function.

Berger (1978) showed some results for losses which are polynomial in thecoordinates of$(\theta-\theta)$

for the normal case, and Brandwein and Strarvderman(1980) for the spherically symmetric distribution when the loss is a nondecreasing concave function of quadratic loss. Here we also study a special form of$He\iota\cdot gers$

loss function for the uniform distribution.

In Section 2, sorne sufficient conditions on the constants a andestimator$\delta_{1}$to dominate

$X$

$b$

for theare given. Further the constants in another$s_{i,\grave{j}}$estimator(1.2)$\delta_{2}(X)=\{1-$

where

$B$

is $pxp$ projection matIix and I is the identity matrix, are determined.

In Section 3, the results in Section 2 are applied to three distributions. The distributions are the truncated normal, the parabola and the cusp shaped distributions which are defined by (3.1),(3.3) and (3.5), respectively. The cusp shaped

distribution is the distribution of the best invariant estimator of the location

parameter of the uniform distribution,
In Section 4, the values of a and$b$for the estimator

to dominate $X$ are$\delta_{1}$given when the distributions of components are rnixture of two uniform distributions with a common center.

In Section 5, we give sufficient conditions on the constants a andestimator$\delta_{1}$to dominate $X$ under the loss

$L(\delta_{1},\theta)=||\delta_{1}-\theta||^{4}$$b$

for thewhen the distribu-

tions of components of $X$ are uniform distributions.

a Estimation or location parameters of the distributions wtth ftnite support.

Let $X_{i}(1=1\ldots.,p)$ components of $X=(X_{1}, \ldots, X_{p})’$ , be$p$

independent ran-dom variables from a probability density of the form

(2.1)

$f_{i}(x_{i}-\theta:)=\{\begin{array}{l}f_{i}(|x_{i}-\theta_{i}|)>0.if|x_{i}-\theta_{i}|\leq c.0,othemise\end{array}$which are assumed to have the same varianceInestimating

the$\delta_{l}(X)=(\delta_{11}(X)\ldots..\delta_{1p}(Y))$

. Set

$V$

vectorlocation$Z_{i}=X_{i^{-1}}1_{i}$$\theta=(\theta_{1}\ldots.

under the squared loss functiDn

3

,

and

$EZ_{\mathfrak{i}}=0$\theta_{p})’$.

by

$\prime JU$.

$||\delta_{1}(X)-\theta||^{2}=1=1Z(\delta_{1i}(X)-\theta_{i})^{2}$

some sufficient conditions on the constants a and
risk

$b$

of (1.1) are given such as theis uniformly smaller than $R(X,\theta)$ .

$R(\delta_{1},\theta)=E_{X}||\delta_{J}-\theta\Vert^{2}$$E_{X}$

denotes theexpe ctation $wi$ th $re$ spect to $X$ .
Theore $m2.1$ . Let X $haYe$ a probability density of the form (2.1). Assume the$fo1arrow$

lowing conditions be satisfied for $i=1\ldots.,p$ :(22)$E^{7}Z_{i^{4}}$く$\infty$.

and there exists a constant $d_{i}>0$ such that

(2.3)$| \int_{\tau_{\{}}^{z}\int^{y_{\epsilon_{i}}}(\int_{\vee}^{u_{i}}rtf_{\mathfrak{i}}(t)dt/V+f_{i}(u))dudy|\leq\phi f_{i}(z)$where $V= \int_{-}^{c}qz^{2}f_{i}(z)dz$ . Then the risk ofil$a \geq 6\sum_{\mathfrak{i}=1}d_{l}/p$$R\cdot oof$

(2.4)

and

$0$

く $b\leq$\delta_{1}$.

is uniformly smaller than that of $X$2(p-2)V$.
.

$R(X, \theta)-R(\delta_{1},\theta)=2bB\{\sum_{iarrowarrow 1}\frac{X_{i}(X_{i}-\theta_{1})}{\alpha+|\llcorner\eta|^{2}}-\frac{b|\Psi||^{l}}{2V(a+||X||^{2})^{Z}}\}$

The conditional expectation of a term of the summation in (2.4) equals

Integration by parts gives(2.5)$I=- \int_{-\tau^{l_{l}}}^{c}g_{1}(\wedge)\int_{-\tau}^{z_{i_{i}}}yf_{i}(y)\ovalbox{\tt\small REJECT} dz_{i}$$=VE_{Z_{l}}\}g_{1}(Z)\}+I_{1}$

.

where

(2.6)

$g_{1}(z_{i})= \frac{1}{a+||z+\theta||^{2}}-\frac{2(z_{i}+\theta_{i})^{2}}{(a+||z+\theta||^{2})^{2}}$and$\varphi$$5_{\perp}$$I_{1}=-V \int^{c_{c^{i}}}g_{1}(z_{i})(\int^{z_{c^{i_{1}}}}yf_{\mathfrak{i}}(y)dy/V+f_{i}(x_{i}))dz_{i}$Integration by parts is applied to(2.7)$I_{1}$.

twice. then$I_{1}=6V \int_{-\tau}^{c_{i_{i}}}g_{2}(z_{i})\{\int_{-c^{1}}^{z}\int_{-\epsilon}^{y}(\int_{-T}^{u_{i}}ff_{i}(t)dt/V+f_{i}(u))dudy\}dz_{i}$

,

where
.

$g_{2}(z_{i})= \frac{1}{(a+||z+\theta|\ovalbox{\tt\small REJECT})^{2}}\frac{8(z_{t}+\theta_{i})^{2}}{(a+||z+\theta||^{2})^{3}}+\frac{8(z_{i}+\theta_{i})^{4}}{(a+||z+\theta||^{2})^{4}}$

Note

that

$\int_{-\tau}^{w_{t}}(\int_{-\tau}^{u_{i}}tf_{i}(t)dt/V+f_{i}(u))du$$\int_{-c_{i}}^{l}\int_{-t}^{y_{\{}}(\int_{\eta}^{u}tf_{i}(t)dt/V+f_{\mathfrak{i}}(u))dudy$

The inequality

isan odd function and thatis an even function.

$|g_{2}(z_{i})| \leq\frac{1}{(a+\Vert z+\theta\Vert^{2})^{2}}$and the condition (2.3) show that

(2.8)

and from the last expression of (2.5) and (2.8),

Hence from (2.4),$R(X,\theta)arrow R(\delta_{1},\theta)$$\geq 2bV$

which iS nonnegatiVe if $a \geq 6\sum_{iarrow}4/p$ and$0$

く $b\leq 2\mathfrak{c}_{P}-2)V$Remark 2.1. Note that $- \int_{-0}^{y}tf_{i}(t)dt/V=f_{i}(y)$ in the normal case. It is shown,by using integration by parts also, that$- \int_{\tau^{t_{l}}}^{c}\int_{q}^{z}\int^{\nu_{c_{t}}}(\int_{-c}^{u_{2}}tf_{i}(t)dt/V+f_{i}(u))aeudydz=\{SZ_{i}^{4}-3(EZ_{i}^{2})^{2}\}/6EZ_{i}^{2}$$\zeta$.

$\tau J_{t}$,Under a stronger condition an alternative sufficient condition is obtained.
The $0\tau em2.2$ . Let

satisfy the conditions of Theorem 2.1 and$f_{\mathfrak{i}}$(2.9)$\int_{-\mathfrak{r}_{1}}^{z}\int_{-e}^{y_{i}}(\int_{arrow\epsilon}^{u}tf_{i}(t)dt\prime V+f_{i}(u))dvdy\geq 0$

Then

the

,

ri $sk$$a\geq 24(2-\sqrt{2})$of$\delta_{1}$uniformly

is$|z|\leq c_{i}$

.

than

smaller

that

of

$X$

if

ma.x $d_{;}/p$ and $0<b\leq 2(parrow 2)V$
$1\leq\Rightarrow$$Proot$ .

The inequality.

$g_{2}(z_{i})\geq-4(2-\sqrt{2})(z_{i}+\theta_{i})^{2}/(a+\Vert z+\theta||^{2})^{8}$(2.7) of Theorem 2.1 and the condition (2.9) give
(2.10)$I_{1}\geq-24(2-\sqrt{2})VE_{4}$‘ $d_{\ddagger}X^{2}/(a+||X||^{2})^{3}$}.

Therefore from the last expression of (25) of Theorem 21 and (2.10),$R(X,\theta)-R(\delta_{1}.\theta)\geq 0$

if the conditions on a and$b$are satisfled.

In Theorem 2.1 and 2.2 it is shown that the estimator
towards the$0\tau ig\dot{m}$,dominates $X$. Here the estimator

estimators towards a sub-space spanned by$B$

sufficient cond\’itions on the constants a and$b$$R(X,\theta)-R(\delta_{l},\theta)is$$?\hslash e$ore $m2.3$ . Let$f_{i}$$X_{i}s$of (1.2), which pulls theis considered, and someof (1.2) are given such as

satisfy the conditions of lheorem 2.1. ‘Ihen the $ri$ sk of$a \geq\frac{6\{8\sum_{iarrowarrow 1}b_{\ddot{\mathfrak{n}}}d_{\ddagger}+\sum_{i=1}(1-b_{\ddot{r}})^{2}4I}{\sum_{i-\urcorner}(1-b_{\ddot{\mathfrak{n}}})}$

and.

く $b \leq 2(p-\sum_{i-\urcorner}b_{\ddot{B}}-2)Y$

whe re $B=(b_{ij})$ .

. vrhich pulls

nonnegative.

uniformly smaller than that of $X$ if$0$,$\delta_{1}$$\delta_{1}$$\delta_{Z}$is5:
Pro a . Theorem 2.3 can be proved in a similar way as Theorem 2.1 and the proof$f$

is omitted.

$7he$

ore $m2.4$ . Let$f_{i}$

satisfy the conditions of Theorern 2.2. Then the risk of$\delta_{2}$isuniformly smaller than that of $X$ ifa$\geq 6_{i-\urcorner i^{\frac{f^{\urcorner}}{=}}i}g_{(1-b_{ii})^{2_{\zeta}}1/(1-}b_{i}i$)and$0<b \leq 2(p-\sum_{=i1}b_{\ddot{u}}-2)V$

Corouaru 2.5. Le$tf_{i}$satisfy the conditions of Theorem 22. Assune that the diag-

onal elements of $B$ are equal. Then the risk of$\delta_{2}$is uniformly snaller than thatof $X$ if $a \geq 24(2-\sqrt{2})\max_{i}d_{\iota}/p$ and $0<b\leq 2\{p(1-b_{\ddot{u}})-2\}V$

Remark 2.2.

[ $fB=(b_{ij}),$ $b_{ij}=1/p$ for all $i$ and $j$ ,estimators towards their average$\overline{X}=$the estimator$\delta_{8}$pulls the$(X_{1}+ +*)/p$ .

a Examples
In this section Theorem 2.1 and Theorem 2.2 ar applied to the truncated$e$normal, the parabola and the cusp shaped distributions to obtain the estimator$\delta_{1}$which dominates $X$ .

$Ex\alpha m_{-}vle3.1$(3.1). Suppose that$z_{i}$

has the common density of the form

$f_{1}(z)= \frac{1}{\sqrt{2\pi}}e^{arrow z^{8}/a_{[[-T.C](Z}})/(\Phi(c)-\Phi(-c))$.

where th is a c.d. . of standard normal distribution and$f$$I_{[\cdot\tau.\epsilon]}(z)$.

is an indicatorfunction ol the interval [-c, ]. (i.e. $I_{[-\tau.c]}(z)=1$ if 1 $z|\leq c$ and $I_{[r.]}(z)=0$ if$c$$|z|>c.)$7$5_{r}$Then, integ ration by parts shows that$p_{1}(z_{i}) \equiv-\int_{\tau^{\{}}^{z}\int_{-r}^{y}(\int_{\sim_{\tau}}^{u}tf_{1}(t)dt/V+f_{1}(u))dvdy$$= \frac{e^{-\epsilon^{2}/2}}{(2\Phi(c)-1)V}q_{1}(z_{i})J_{1}(\approx)$

.

where
$q_{1}(z_{i})=\{2cz_{i}\Phi(z_{i})+2c^{2}\Phi(-c)-(z_{i^{2}}-c^{2})(\mathfrak{X}(c)-1)/2$$-c(z_{i}+c)\}ez_{\mathfrak{i}^{2}}+2c(1-e)/\sqrt{2\pi}$ .

Note

that

for

$q_{1}(z_{i})\leq 0$$0\leq z_{i}\leq c$.

The$inequalit_{Y}$$\Phi(z_{i})\geq(2\Phi(c)-1)z_{i}/2c+1/2$ gives$-q_{1}(*)\leq(-k_{1}z_{i}^{2}/2+c^{2}k_{1/2}+k_{2})e^{z_{i^{2}}/2}-2c/\sqrt{2\pi}$where $k_{1}=2\Phi(c)-1$ and(3.2)$k_{2}=2c\epsilon^{-\tau^{g/2}}/\sqrt{2\pi}$for $D\leq z_{i}\leq c$ ,. and$-p_{1}(z_{i})\leq(k_{1}e^{arrow 1+k_{8}/k_{1}}-k_{8})f_{1_{(}}(z_{i})/k_{1}V$

By Tbeorem 2.2 and (3.2), the risk of$a\geq a_{0}$. where$\delta_{1}$is uniformly smaller than that of $X$ if$\alpha_{0}=24(2-3(k_{1}e^{\neg}+k_{8}/k_{1}-k_{2})/k_{1}pV$

and$0<b\leq 2(p-2)V$,

Table 3.1 gives the value of$a_{0}$

for $c=1$ ; $c=2$ andTable 3.1

$Exam_{-}vte3.2$(3.3). Suppose that

$Z_{i}$$a_{0}$.

: the lower bound of a

has the common density of the form

$f_{2}(z)= \frac{3}{4c^{l}}(c^{2}-z^{2})I_{[- c.\epsilon[}(z)$.

Without loss of enerality we set $c=1$ . Then$g$$\acute{c}=\infty$

53
(3.4)

$- \int_{-1^{i}}^{z}\int_{\neg}^{y}(\int_{\neg}^{u}tf_{2}(t)(It/V+f_{2}(u))dudy=-\frac{1}{24}(z_{\iota^{2}}-1)^{2}f_{2}(z_{i})$$\geq-f_{2}(z_{i})/24$

According to Theorem 2.2 and (3.4),$a\geq 5(2-\sqrt{2})V/p$$Exam\mathcal{D}le3.3$

(3.5)

and

$0$

$R(X,\theta)-R(\delta_{1},\theta)$.

is nonnegative if

く $b\leq 2\Phi-2)V$. Suppose that

$Z_{i}$

has the common density of the form

$f_{3}(z)= \frac{k+1}{2c^{k+1}}(c-|z|)^{k}I_{[-\sigma.c]}(z)$.

where $k\geq 0$ . Assume $c=1$ . Then$p_{3}(z_{i} I\equiv-\int_{\neg}^{z}\int_{\neg}^{y}(f_{\neg}^{u}tf_{3}(t)dt/V+f_{3}(\tau\iota))d\tau\iota dy$

for $0\leq z_{i}\leq 1$$= \frac{(1-z_{i})^{2}}{2(k+1)(k+2)(k+4)}q_{3}(z_{i})f_{3}(z_{i})$ ,where$p_{3}(z_{i}),$$q_{S}(z_{i})=-(k+1)(k+2)z_{i^{2}}+(k+2)(k-2)z_{i}+k-2$.
$ve$

To

.
evaluateconsider the following two cases: $0\leq k\leq 2$ and $k>2$ .

Case 1: $0\leq k\leq 2$ . Noting that $q_{3}(z_{i})\leq 0$ , we can show that
(3.6)$-p_{3}(z_{i})\leq l_{1}(l_{2}+l_{3}l_{4^{1/2}})Vf_{3}(z_{\dot{t}})$

,

where
$l_{1}=(k+3)/204B(k+1)^{4}(k+2)^{8}(k+4)$.

$l_{2}=arrow 27k^{6}-252k^{5}-620k^{4}+640k^{S}+4992k^{2}+716Sk+1024$ ,$l_{3}=\Re^{4}+62k^{3}+72k^{2}-96k+128$ ,and$l_{4}=(k+2)^{l}(5k-4)^{2}-16k(k-2)(k+1)(k+2)$ .

Therefore

from

Theorem$a \geq 24(2-\sqrt 2\gamma\iota_{1}(l_{2}+l\oint_{4}^{1/2})V/p$

2.2

and

(3.6),$R(X,\theta)-R(\delta_{1},\theta)\geq D$

if

and $0<b\leq 2(p-2)V$

Case 2: $k>2$ . Noting that$q_{3}(z_{i})\{\geq 0$$otherTb \dot{2}^{\dot{l}}se1f0\leq z\leq.\frac{(k+2)(k-2)+\{(k+2)^{2}(k-2)^{2}+4(t-2)(k+1)(k+2)\iota^{\iota/\epsilon}}{2(k+1)(k+2)}$

く $0$ ,$o/$,

$S\cup$

we get.

$\{p_{3}(z_{l})|\leq l_{1}(|l_{2}|+l_{3}l_{4}^{1/2})Vf_{3}(z_{i})$

(3.7)
By
$0$

Theorem 2.1 and (3.7),$a\geq 6l_{1}(|l_{2}|+l_{3}l_{4}^{1/2})Y$if

2(p-2)V$.

く $b\leq

Remark$R(X,\theta)-R(\delta_{1}.\theta)\geq 0$of the tocationma
es timator
3.1. Note that the best invariant

density (3.5).
the probability$-the$
uniform distribution has
33, the estimator$\delta_{1}$has a smaller risk thanvariance. Let
In thiscase where$Z_{i}$

distributions,

$whe$

re

$\dot{\iota}e$are obtained by applying

2. 1.

of the estimator

in]Theorem 2.1. Here we$\delta_{1}$study theis a mixture of two uniform$Z_{i}$

have,$(z)+ \frac{1-a}{2c_{2}}t_{[-c_{2}.c_{8}}(z)$く 1 and $c_{1}\leq Cg$

orem41.

$b$. the density is$f(z)= \frac{a}{2c_{1}}I_{[}-r_{1}.c_{1}$$0\leq\alpha$with the same(2.2) and (2.3) of Theorem

density which
has the commOn$?h\underline{e}$ Let$\delta_{1}$of uniform distributions.

on the constants a andcase, sufficient conditions$X$

According to Example

have mixture of distributionSes satisfy the conditionS
their densities(1.1) to dominate

(4.1)

$X$

parameter ofthe best invariant one.

parameters for the mixture
4. Estimation of locationofSuppose that componentSand.

the form (4.1).
the common density ofis unifornly smaller than

Then the risk ofthat of $X$ if$a\geq rnax\{6c_{0}+3(zarrow\sqrt{2})c5/P, 3c_{Q}+(9c_{0}^{Z}+60c_{0}(2arrow\vee\overline{2})c_{2}^{2})^{1/2}\}$

and$0<b\leq 2(p-2)V$

.

vhere

$/\theta$5$V_{c_{1}}= \int_{-\tau_{t}^{l}}^{c}\frac{z^{2}}{2c_{i}}\ovalbox{\tt\small REJECT}$,and$V= \int_{-\tau^{i_{i}}}^{c}z^{2}f(z)\ =aV_{1}+(1-a)V_{c_{3}}$$f$.

IYote. It is easily seen that$c_{0}=\{\begin{array}{l}\frac{a(1-a)_{C_{1}C}dc_{2}-c_{1})(V_{c_{2}}-V_{c_{1}})}{2((1-a)c_{1}+ac_{Z})V\prime}if\alpha\leq\frac{c_{1}^{I}}{c_{2}(c_{2}-c_{1})+c_{1}^{2}}\frac{a(c_{2}-c_{1})^{2}(V_{c_{l}}arrow V_{c_{1}})}{2V}ifa>\frac{c_{1}^{2}}{c_{2}(c_{l}-c_{l})+c_{1}^{2}}\end{array}$ffoof. From (2.4) of Theorem 2.1, the conditional expectation of a term of thesumvnation in $\{R(X,\theta)-R(\delta_{1},\theta)\}/2b$ can be written as$L \equiv E_{\eta}\{\frac{X_{i}(X_{i}arrow\theta_{2^{i}})a+||X||}{}$$=a \int_{-c^{1_{1}}}^{c}\frac{z_{i}(z_{i}+\theta_{i})}{a+\Vert z+\theta\Vert^{2}}\frac{1}{2c}arrow Z_{i}1+(1-\alpha)\int_{2}^{c_{2}}-\frac{z_{i}(z_{i}+\theta_{i})}{a+||z+\theta||^{2}}\frac{1}{2c_{2}}dz_{i}$Integration by parts shows that
(4.2)

$f_{-\tau^{i_{j}}}^{z} \int_{\tau}^{r_{i}}(\int_{arrow}^{u_{j}}tdt/2c_{j}V_{\epsilon_{f}}+1/2c_{j})\phi\iota\otimes$$= \frac{1}{16c_{j}^{3}}(z_{i^{2}}-c_{j}^{a})^{2}j=1$,Z.

The last expression of (2.5) of Theorem 2.1 and (4.2) give
(4.3)

$L \geq\alpha V_{\epsilon}J_{-r^{1_{1}}}^{c}\frac{f_{1}(z_{i})}{2c_{1}}z_{i}+(1-\alpha)V_{c_{I}}\int_{-}^{c_{8_{2}}}\frac{t_{1}(z_{i})}{2c_{2}}z_{i}$$=VE_{Z_{i}}\{t_{1}(Z_{i})\}+L_{1}$.

schere.

$t_{1}(z_{i})= \frac{1}{a+||z+\theta\Vert^{z}}-\frac{2(z_{i}+\theta_{i})^{2}}{(\alpha+||z+\theta||^{2})^{2}}-\frac{3(2-\sqrt 2\urcorner c_{2^{2}}(z_{i}+\theta_{i})^{2}}{(a+||z+\theta\Vert^{2})^{3}}$$L_{1}=V \int_{\tau^{2_{l}}}^{c}t_{1}(z_{i})(\tilde{f}(z_{i})-f(z_{i}))dz_{i}$,an d$\tilde{f}(z)=\frac{\alpha V_{c_{l}}}{2c_{1}V}I_{[-_{1}.c_{1}]}(z)+\frac{(1-\alpha)V_{c_{8}}}{2c_{2}V}I_{[\neg_{g}.cd}(z)$Note that $f(z\sim)-1(z)>0$ if $C1$ く $|Z|\leq c_{2}$

//

. and

.

$\tilde{f}(z)-J(z)$ く $0$

if $|Z|\leq c_{1}$$\mathfrak{c}J\cup$Integration by parts is applied totwice, then$L_{1}$$L_{1}=-6V \int_{\neg}^{c_{2_{2}}}t_{2}(z_{i})\tilde{g}(z_{i})dz_{i}$

.

where

$t_{2}(z_{i})= \frac{1}{(a+||z+\theta||^{l})^{2}}+\frac{(2-\sqrt{2})c_{Z^{2}}-8(z_{i}+\theta_{i})^{2}}{(a+||z+\theta\Vert^{2})^{3}}$$+ \frac{8(a+\theta_{i})^{4}-15(2-\sqrt{2})c_{\epsilon}^{2}(z_{i}+\theta_{i})^{2}}{(a+\Vert z+\theta||\mathfrak{J}^{4}}$$+ \frac{24(2-\sqrt{2})c\S(z_{i}+\theta_{i})^{2}}{(a+||z+\theta\Vert^{2})^{6}}$and$\tilde{g}\{z_{i})=(1arrow\alpha)(V_{c}\swarrow Varrow 1)z_{\mathfrak{i}^{2}}/4c_{2}-(1-\alpha)(V_{c}\swarrow V-1)|z_{i}|/2$$+(1-\alpha)(V_{c_{2}}/V-1)c_{2}/4$ ,if$c_{1}$く$|z_{i}|\leq c_{2}$.

$=a(V_{e_{1}}/V-1)z_{i^{\S}}/4c_{1}+(1-\alpha)(V_{c_{8}}/Y-1)z_{i^{2}}/4c_{2}$$+(1-\alpha)(V_{c_{S}}/V-1)(c_{2}-c_{1})/4$.

if$|z_{i}|\leq c_{1}$.

/Note thatis positive, eyen and umimodal on $(-c_{2},c_{2})$ and $f(z_{i})$ is a step$\tilde{g}(z_{i})$function of the same property. Thus,$z_{\ddagger}=0mz_{i}=c_{1}$ ,(4.4)$\tilde{g}(z_{1})\leq c$$\tilde{g}(\approx)/f(z_{i})$has the maximurn value when

and

of

$(z_{i})$.

The inequalities (4.4) and$t_{2}(z_{i})\leq 1/(a+||z+\theta||^{2})^{2}-10(2arrow\sqrt{2})c_{I}^{2}/(a+\Vert z+\theta\Vert^{2})^{3}$

gve
(4.5)

Therefore from (4.3) and (4.5),$\sqrt 2$.

$5_{i}\grave{j}$Hence$R\{X,\theta)-R(\delta_{1},\theta)$$\geq 2bVE\{\frac{ap+(p-2)||X||^{2}-6c_{Q}p-3(2-\sqrt{2})c_{2^{2}}}{(a+|[X||^{2})^{2}}$$+ \frac{3(2-\eta 2ac_{2^{2}}-6(Xs-\sqrt{2})c_{0}c_{2}*}{(a+||X||^{2})^{3}}-\ovalbox{\tt\small REJECT}_{2V(a^{b}+}^{x_{||X||^{2}}^{2}})^{2}$

which is nonnegative if$a \geq\max\{6c_{0}+3(2-\sqrt{2})c_{2}^{2}/p , 3c_{0}+(9c_{0^{2}}+60c_{0}(2-\sqrt{2})c_{2^{8}})^{1/2}\}$.

and$0<b\leq 2(p-2)V$

Table 4.1 shows the lower bound of $a/V$ for $p=3;c_{2}/c_{1}=1.5,2,2.5$ and$a=0.2,0.4,0.6,0.8$ . The lower bound of $a/V$ becomes large when$c_{2}/c_{1}$

or

$a$

is

large.

Table 4.1 The lower bound of $a/V$

Remark 4.1 Theorem 4.1 can be extended to the$m\dot{\iota}xture$of n-uniform distribu-

tions.

a Estimation of location parameLers for the uniform distribution under theloss$||\delta_{1}-\theta||^{4}$

In this section the constants a andthe estimator$\delta_{1}$$b$of (1.1) are given such as the risk ofis uniformly smaller than that of $X$ with respect to the loss2

(5. 1)$|| \delta_{1}-\theta||^{4}=\sum_{-\urcorner}(\delta_{1i}-\theta_{i})^{2}$$/J$

$\hat{\iota)}\cup$for the uniform distribution.

Following Herger$s$notation (1978). let$\gamma(X)=(\gamma_{1}(X), \ldots, \gamma_{p}(X))$$=\delta_{1}(X)-X$

and$0or_{\mathcal{R}}iif^{r}i=0,$く$c_{n.i}=\{l101,-\epsilon$$if2\leq i\leq ifi$

:

$g$

is

$R^{1}arrow R^{1}$

$n$

$h$

:

$R^{p}arrow R^{1}$$i$

is odd,times differentiable, let$g^{(i)}(z)= \frac{d^{i}}{dz^{j}}g(z),$

If

or

.

,

If

$>\tau\iota$$0\leq j\leq n,$$(g^{(0)}(z)=g(z))$ .

is a function with sufficient order derivatives, let$h^{i(k).j(1)}(z)= \frac{\partial^{(k*)}}{\partial z_{i^{k}}\partial z_{j^{l}}}h(z)$.

The following lemma is useful in carrying out integration by parts for thelo$ss||\delta_{1}-\theta|_{t}^{I4}$.

Lemma 5.1. Let $Z$ have the density of the forn$f(z)=\{\begin{array}{l}f(|\iota|)_{l}if|z|\leq c0,othe\tau?\underline{1}\dot{n}se\end{array}$Suppose that $g(x)$ is a real-valued$n$tirnes continuously differentiable function

and that$f_{arrow}^{c_{C}}|g^{(n}7)_{(Z)|}|z|^{j}f_{n\urcorner}\cdot(z)dz$

く$\infty$

for $0\leq j\leq n$ ,whe re$f_{i}(z)= \int_{\tau}^{z}-yf_{i-}1(y)dy$

for $1\leq i\leq n$ and $f_{0}(z)=f(z)$ .

Then$E(g(Z)Z^{n} \}=\int_{-}^{c}\tau_{larrow}\sum_{arrow 0}^{n}c_{n.i}g^{(\hslash^{-i)}}(z)f_{[n}$$/\not\in$

ti $/z$ ] $(z)dz$.

$5_{\perp}$

where $[]$ denotes Gauss’ $sy$ rnbol.

Froo $f.$ This lemma can be proved by induction on$n$along the same line as in the

proof of Lemma 1 in Berger (1978).
Theore $m5.2$ . Let $Z_{i}(i=1,\ldots.p)$ be$p$independent random variables from aform distribution on $(arrow c,c)$ . Then the risk of the es timator$\delta_{1}$$uniarrow$is uniformly smaller

than that of $X$ with respect to the loss given by (5. 1) if$a \geq\frac{p+107/30}{p+4/5}b+\frac{33L}{20}V$and

Pro $of$ .
$| I\delta_{1}-\theta||^{4}=|_{i\urcorner}\sum_{arrow}(\gamma_{i}(x)+\eta-\theta_{\mathfrak{i}})^{2}\}^{2}=$.

$\{\sum_{i\Xi}(\gamma_{i}(z+\theta)+z_{\iota})^{2}\}^{2}$

Iherefore(5.2)$\Delta(\theta)=E||\delta_{1}arrow\theta||^{4}-E||X-\theta||^{4}$$=E \{4\sum_{i}\sum_{j}\gamma_{i}\gamma_{j}Z_{\iota}Z_{j}+\sum_{i}\sum_{j}\gamma_{i^{2}}\gamma_{j}^{2}+4\sum_{i}\sum_{j}\gamma_{i}Z_{1}Z_{j}^{2}$$+2 \sum_{i}\sum_{j}\gamma_{i^{2}}Z_{j}^{z}+4\sum_{i}\sum_{j}\gamma_{i}\gamma_{j}^{2}Z\}$.

Using Lemma 5.1 to evaluate the conditional expectation of a term of each summation in (5.2), we get$E_{2_{l}} \{\gamma_{i^{l}}Z_{i}^{2}\}=2\int_{-\tau}^{c}\{(\gamma_{i^{i(I)}})^{2}+\gamma_{i}\gamma_{i^{\mathfrak{i}(2)}}\}f_{Z}(z_{i})dz_{j}+\int_{\tau}^{c}\sim f_{i}f_{1}(z_{i})dz_{i}$.

$E_{4^{Z_{j}},} \{\gamma_{i}\gamma_{j}Z_{i}Z_{J^{\vee}}\}=f_{-\tau}^{c}\int^{c_{c}}\{\gamma_{i^{i(1).j(1\}}}\gamma;+\gamma_{i}^{i(1)}\gamma_{J^{j(1)}}$$+\gamma_{1}^{j(1)}\gamma_{j^{i(1|}}+\gamma_{i}\gamma_{j^{i(1).j(1)}}\}f_{1}(z_{\dot{t}})f_{1}(z_{j})dz_{i}dz_{j}$(5.3)$E_{Z_{i}} \{\gamma_{i}Z_{i^{3}}\}=\int_{-c}^{c}\gamma_{i^{i(3|}}f_{3}(z_{i})dz_{i}+3\int_{-\tau}^{\epsilon}\gamma_{i^{\mathfrak{i}(1)}}f_{2}(z_{i})dzi$,,$E_{Z_{t}.Z_{j}}| \gamma_{i}Z_{i}Z_{j}^{2}\}=\int_{-c}^{c}\int_{-\tau}^{c}\gamma_{i^{i(1).j(2)}}f_{1}(z_{i}\rangle J_{2}(z_{j})dz_{i}dz_{j}$$+ \int^{c_{c}}\int_{\neg}^{c}\gamma_{i^{i(1)}}f_{1}(z_{i})f_{1}(z_{j})dz_{i}dzj$,$E_{Z_{f}} \{\gamma_{i}^{2}Z_{j}^{2}\}=2\int_{-\mathfrak{r}}^{c}\{(\gamma_{t}^{j(1)})^{2}+\gamma_{i}\gamma_{l}^{j(2)}\}f_{2}(z_{j})dz_{j}+\int_{\tau}\gamma_{i^{2}}f_{1}(z_{j})dz_{j\prime}$$/\zeta$6.
and$E_{Z_{l}} \{\gamma_{i}\gamma_{\dot{\gamma}^{2}}Z_{i}\}=\int_{-}^{c_{C}}\{\gamma_{i^{i(1)}}\gamma_{j^{2}}+\eta_{\mathfrak{i}}\gamma_{j}\gamma_{j^{i(1)}}\}f_{1}(z_{i})dz_{i}$

,

where

$\gamma_{i}(z+\theta)=-\frac{b(z_{i}+\theta_{i})}{a+||z+\theta||^{2}}$$f_{2}(z_{\mathfrak{i}})= \frac{1}{18c}(z_{i^{I}}-c^{I})^{I}$,and$f_{1}(z_{i})= \frac{1}{4c}(arrow z_{i^{2}}+c^{2})$.

$f_{3}(z_{t})= \frac{1}{96c}(-z_{i^{l}}+c^{2})^{3}$.

S\’imilarly in the proof of Theorem 2.1, a straightforward calculation of each termin (5.3) gives
(5.4)

where
$m_{0}=-4p(p+4/5)c^{4}/9$,$m_{1}=2(p+2)c^{2}b/3+8(p+4/5)c^{4}/9$ ,$m_{2}=4p(p+107/30)c^{4}b/9+61p(p+89/61)c^{8}/135$,$m_{3}=-4\{p+2)c^{2}b^{2}/3-176\phi+19/11)c^{4}b/45-128\otimes+2)c^{g}/135$$m_{4}=pc^{s}b^{2}+4p(p-1/10)c^{6}b/3+\psi(g-1)c^{8}/5$$m_{5}=b^{3}+ae^{Z}b^{2}+64c^{4}b/5+128c^{6}/45$

.

.

.

.

$m_{6}=21\phiarrow 4$ノ $7)c^{4}b^{8}/5+1492[p-1477/1492)c^{8}b/75+186(p-1)c^{8}/25$$m_{7}=3p\{p-1$ )$c^{8}b$ノ 2,$m_{8}=352\zeta_{P}-1$)$c^{8}b$,

and $m_{9}=480c^{6}b$.

Befine$v(y. \alpha.b)=m_{0}+\frac{m_{1}y+m_{2}}{a+y}+\frac{m_{S}y+m_{4}}{(a+y)^{2}}+\frac{m_{5}y^{2}+m_{6}y+m_{?}}{(a+y)^{3}}$$+ \frac{m_{8}y}{(acdot y)^{4}}+\frac{m_{9}y^{l}}{(a+y)^{5}}$It is clear from (5.4) that$\Delta(\theta)\leq 0$

if a and$b$

can be chosen so that $v(y,a,b)\leq 0$

for all $0\leq y\leq\infty$ . To evaluate $v(y,a,b)$ , we decompose $v(y,\alpha.b)$ as$v(y,a,b)\equiv v_{l}(y,a,b)+v_{2}(y,a,b)$ ,$t_{J_{-J}}$

where$v_{1}(y, \alpha,b)=m_{0}+\frac{m_{1}y+m_{e}}{a+y}+\frac{m_{4}}{(a+y)^{2}}+\frac{m_{6}’y+m_{7}}{(a+y)^{3}}+\frac{m_{8}y}{(a+y)^{4}}$$v_{2}(y,a.b)= \frac{m_{3}y}{(a+y)^{2}}+\frac{m_{5}y^{2}+m_{6}’’y}{(a+y)^{3}}+\frac{m_{9}y^{2}}{(a+y)^{5}}$$m_{6}’=3(p-2/5)c^{4}b^{2}+1492\otimes-1477/1492)c^{6}b/45+186(parrow 1)c^{8}/25$ ,and$m_{6}’’=6(p-1)c^{4}b^{2}/5$ .

It is straightforward to verify that $v_{1}(y.\alpha,b)\leq 0$ and $v_{2}(y,a,b)\leq 0$ if theconditions on a and$b$are satisfied. The proof is completed.

Acknowle dgement
The author would like to thank Prof. M. Sibuya, Prof. Y. Washio and Dr. N. Shinozaki of Keio University for their helpful suggestions and criticisms.

References[1] Berger, J. (1978). Minimax estirnation of multivariate normal mean underpolynomial loss. J.Multivariate Anal. 8, 173- 180[2] Brandwein, A. C. and Strawderman, W. E. (1980). Minimax estimation of location parameters for spherically symmetric distributions with concave loss.

Ann. Statist. 8279-284[3] Brown, L. D. (1966). On the admissibility of invariant estimators of one ormore Location parameters. Ann. MMath.-Statist. 37 $1087arrow 1136$[4] Hudson, M. (1978). A natural idenbty for exponential families
tions in multiparameter estimation. Ann. Statist 6473-484/7$\backslash rith$applica-S.
[5] James, W. and Stein, C. (1961). Estimation with quadratic loss. Proc. FourthBerkeley Symp. Math. Statist. Prob. 1381-379 Uni . ($f$ California Press.
$v$[6] Shinozaki, N. (1984). Simultaneous estimation of location parameters underquadratic los . Ann. Statist. 12 $322arrow 335$$s$[7] Stein, C. (1955). Inadmissibility of the usual estimator for the mean of amultivariate normal distribution. Proc. Third Berkley Symp. Math Statis .
$t$Prob. 1197-206 Univ. of California Press.
[B] Stein, C. (1973). Estimation of the mean of a multivariate normal distribu-

tion. Proc. Prague Symp. Asymptotic Statist. 345-381$/^{r}K$