Admissibility and minimaxity of generalized Bayes estimators for spherically symmetric family
We give a sufficient condition for admissibility of generalized Bayes estimators of the location vector of spherically symmetric distribution under squared error loss. Compared to the known results for the multivariate normal case, our sufficient condition is very tight and is close to being a necessary condition. In particular we establish the admissibility of generalized Bayes estimators with respect to the harmonic prior and priors with slightly heavier tail than the harmonic prior. We use the theory of regularly varying functions to construct a sequence of smooth proper priors approaching an improper prior fast enough for establishing the admissibility. We also discuss conditions of minimaxity of the generalized Bayes estimator with respect to the harmonic prior.
arxiv-admissibility.tex
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\begin{document}

\begin{frontmatter}
\title{
Admissibility and minimaxity of generalized Bayes estimators for
spherically symmetric family}
\runtitle{Admissibility for spherical family}
%\thankstext{T1}{Footnote to the title with the `thankstext' command.}

\begin{aug}
\author{\fnms{Yuzo} \snm{Maruyama}
 \thanksref{t1}
\ead[label=e1]
{maruyama@csis.u-tokyo.ac.jp}
}
and
\author{\fnms{Akimichi} \snm{Takemura}
% \thanksref{t3}
\ead[label=e2]
{takemura@stat.t.u-tokyo.ac.jp}
}
%\and
%\author{\fnms{Third} \snm{Author}
%\ead[label=e3]{third@somewhere.com}
%\ead[label=u1,url]{www.foo.com}}

\thankstext{t1}{Corresponding author}
%\thankstext{t2}{First supporter of the project}
%\thankstext{t3}{Second supporter of the project}
\runauthor{Y.Maruyama and A.Takemura}

\affiliation{The University of Tokyo}

\address{
Center for Spatial Information Science, \\
The University of Tokyo\\
5-1-5 Kashiwanoha, Kashiwa-shi, Chiba 
277-8568, Japan\\
E-mail: \printead[maruyama@csis.u-tokyo.ac.jp]{e1}}

\address{
Graduate School of Information Science and Technology\\
The University of Tokyo\\
7-3-1 Hongo, Bunkyo-ku, Tokyo 
113-0033, Japan\\
E-mail: \printead[takemura@stat.t.u-tokyo.ac.jp]{e2}}

%\address{Address of the Third author\\
%usually few lines long\\
%usually few lines long\\
%\printead{e3}\\
%\printead{u1}}
\end{aug}

\begin{abstract}
  We give a sufficient condition for admissibility of generalized Bayes
  estimators of the location vector of spherically symmetric distribution
  under squared error loss.  Compared to the known results for the
  multivariate normal case, our sufficient condition is very tight and is
  close to being a necessary condition.  In particular we establish the
  admissibility of generalized Bayes estimators with respect to the
  harmonic prior and priors with slightly heavier tail than the harmonic
  prior.  We use the theory of regularly varying functions to construct a
  sequence of smooth proper priors approaching an improper prior fast
  enough for establishing the admissibility. We also discuss conditions of
  minimaxity of the generalized Bayes estimator with respect to the
  harmonic prior.
\end{abstract}

\begin{keyword}[class=AMS]
\kwd[Primary ]{62C15}
\kwd[; secondary ]{62C20}
\kwd{26A12}
\end{keyword}

\begin{keyword}
\kwd{admissibility}
\kwd{spherically symmetric distribution}
\kwd{minimaxity}
\kwd{regularly varying function}
\kwd{harmonic prior}
\end{keyword}

\end{frontmatter}

\section{Introduction}
\label{sec:intro}
We consider estimation of the $p$-dimensional location parameter of a
spherically symmetric distribution. Specifically, let $ X =
(X_1,\ldots,X_p)' $ have a density function $ f(\|x-\theta \|) $ and
consider estimation of $ \theta $ with a general quadratic loss
function $ L_Q(\theta,\delta)= ( \delta - \theta)'Q( \delta -
\theta)=\|\delta - \theta \|^2_Q $ for a positive definite matrix $Q$.
The usual minimax estimator $ X $, which is generalized Bayes, is
inadmissible for $ p \geq 3 $ as shown in \cite{Stein1956} in the normal
case, and in \cite{Brown1966} under more general situation, respectively.
In the decision-theoretic point of view, we are interested in
proposing admissible estimators dominating $ X $, that is, minimax
admissible estimators.  Note that the dominance over $X$ means
minimaxity in our setting because $ X$ is minimax with a constant
risk.  Note also that our results hold for the location vector of
elliptically-contoured distributions, because we are considering a
general quadratic loss function with arbitrary positive definite $Q$.


In the normal case, there already exists a broad class of admissible
minimax estimators. \cite{Baranchik1970} gave a sufficient condition for
minimaxity of a shrinkage estimator of the form
\begin{equation} 
\label{orthogonal}
 \delta_{\phi}(X)=(1-\phi(\|X\|)\|X\|^{-2})X.
\end{equation}
\cite{Strawderman1971} found a subclass of proper Bayes estimators
of the form \eqref{orthogonal} 
satisfying the sufficient conditions for minimaxity.
\cite{Brown1971} gave a very powerful sufficient condition for
admissibility of generalized Bayes estimators.
Using Brown's (1971) condition, 
\cite{Berger1976b}, \cite{Fourdrinier-etal1998} and
\cite{Maruyama1998, Maruyama2004}
enlarged a class of admissible minimax estimators
which are of the Strawderman type and generalized Bayes. 
For a subclass of scale mixtures of multivariate normal distributions
which includes multivariate-$t$ distribution, 
some proper Bayes minimax estimators were proposed by \cite{Maruyama2003}
by using Strawderman's (1971) techniques.

However, for general spherically symmetric distributions, no minimax
admissible estimators of the location vectors have been derived,
although for the minimaxity, various Baranchik-type sufficient
conditions of the estimator \eqref{orthogonal} were given by
\cite{Berger1975}, 
\cite{Brand-Straw1978, Brand-Straw1991b} and \cite{Bock1985}.  The
main reason is the lack of a standard class of generalized or proper
Bayes estimators of the form \eqref{orthogonal} like the Strawderman
type in the normal case, which allows an easy check of the minimaxity
condition.  Furthermore no sufficient condition for admissibility of
generalized Bayes estimators has been derived.  In 
\cite{Maruyama-Takemura2005}, we 
provided satisfactory solutions to these problems.
In this paper we will weaken 
some regularity conditions assumed in \cite{Maruyama-Takemura2005}.

In Section 2, we give preliminary results including the properties of
regularly varying functions and asymptotic behaviors of expected
values when $\|\theta\|$ is sufficiently large.  The former is useful
for constructing a very convenient sequence of proper densities
%$g(\theta)h_i^\gamma(\theta)$, $i=1,2,\ldots$, 
approaching an improper
density $g(\theta)$, which is required in applying the method of 
\cite{Blyth1951}.  

In Section 3, we will present a powerful sufficient
condition for admissibility of generalized Bayes estimator and in
particular show that the generalized Bayes estimators with respect to
the harmonic prior $g(\theta)=\| \theta \|^{2-p}$ and with respect to 
a prior with a slightly heavier tail 
\begin{align} 
\label{eqn:2-p+log}
g(\theta)= \| \theta \|^{2-p}\log(\|\theta\|+c), \quad c>1, 
\end{align}
are admissible under mild regularity conditions on $f$.  

In Section 4, we show that the generalized Bayes estimator with respect to
the harmonic prior is written as
\begin{equation} \label{harmonic-Bayes}
\left(1-\frac{\int_{0}^{1}t^{p-1}F(t \|X\|)dt}
{\int_{0}^{1}t^{p-3}F(t \|X\|)dt} \right)X,
\end{equation} 
where $ F(u)=\int_{u}^{\infty}sf(s)ds $.  This form is simple enough
to check various sufficient conditions for minimaxity and 
we demonstrate that \eqref{harmonic-Bayes} is minimax for some $f$.
We believe that \eqref{harmonic-Bayes} is 
minimax for a broad subclass of spherically
symmetric distributions.
Notice that the generalized Bayes estimators with respect to
priors except $\|\theta\|^{2-p}$ do not have such simple forms 
as far as we know.

Our proof of admissibility is mainly based on the techniques of
\cite{Brown-Hwang1983}.  \cite{Brown-Hwang1983} considered the problem of
estimating the natural mean vector of an exponential family under a
quadratic loss function.  Note that the intersection of their setting and
our setting is the multivariate normal case.  Their sufficient
condition for admissibility in the normal case does not however permit
$g(\theta)$ to diverge to infinity around the origin like $ \| \theta
\|^{2-p}$,  while it permits $g(\theta) \leq \| \theta\|^a$ with $a\leq 2-p$
for sufficiently large $\|\theta \|$.  Prior to \cite{Brown-Hwang1983},
\cite{Brown1971} considered the estimation in the multivariate normal case
%the problem of
%estimating a multivariate normal mean under a quadratic loss
and gave a powerful sufficient condition for minimaxity which are satisfied
by the harmonic prior and \eqref{eqn:2-p+log}, but his proof was based on many advanced
mathematics. Our mathematical tool is much more familiar to the readers.
\cite{Brown1971} also gave sufficient condition for inadmissibility.  By
using it we see that the generalized Bayes estimator with respect to
$\| \theta \|^{2-p}\log^2 (\|\theta\|+2)$ is inadmissible. Hence our
sufficient condition for admissibility should be very tight and close to
being a necessary condition.  

\cite{Brown1979} considered a more general problem than ours: estimation of
$ \theta $ for a general density $p(x-\theta)$ and a general loss function
$ W(\delta-\theta)$.  He conjectured that the prior $g(\theta) \sim
\|\theta\|^{a}$ with $a\leq 2-p$ for sufficiently large $ \|\theta\|$
leads to admissibility, regardless of the density $p$ and the loss $W$.
% by investigating the asymptotic behavior of the risk function.  
Hence our results support \cite{Brown1979}'s conjecture for the case of
elliptically-contoured family and a general quadratic loss function.

Finally we notice that the most important key for our proof for
admissibility is the construction of a very convenient sequence
$h_i(\theta)$ for approximating $g(\theta)$ by 
$g(\theta)h^\gamma_i(\theta)$ for $0<\gamma \le 2$.
\cite{Brown-Hwang1983} used $ \gamma=2$ and
\begin{align*}
 h_i(\theta)=\begin{cases}
	      1 & \|\theta\| \leq 1 \\
1-\log \|\theta\|/\log i & 1 \leq \| \theta\| \leq i \\
0 & \|\theta\|>i .
	     \end{cases} 
\end{align*} 
This $ h_i(\theta) $ is not differentiable at $ \|\theta\|=1$ and
truncated at $\|\theta\|=i$, which makes handling and extension difficult
for our purposes.  Our $h_i(\theta)$ given in Section 2 is smoother and
not truncated. 
Furthermore our $\gamma$ is flexible whereas $\gamma=2$ in 
\cite{Brown-Hwang1983}. By such a flexible $\gamma$, 
we can adjust the rate of convergence of
$ g(\theta)h^\gamma_i(\theta) $ so that it is just enough to be proper.
We will see that choosing as small $\gamma$ as possible is
important in the main theorem, Theorem 3.1.
We believe that our smooth function $h_i(\theta)$ and an idea of 
flexible $\gamma$ are very useful for showing
admissibility of generalized Bayes estimators in various problems.


\section{Preliminaries}
\label{sec:preliminaries}
In this section we prepare a sequence of proper densities using the
theory of regularly varying functions and give some results on
asymptotic behaviors of expected values when the location parameter
diverges to infinity.  For the theory of regularly varying and slowly
varying functions the readers are referred to \cite{geluk-dehaan1987}
and \cite{BGT1987}.

\subsection{Regularly varying functions}
\label{sec:rv}
A Lebesgue measurable function $ f: R^+ \to R $ which is eventually
positive is called regularly varying if for some $\alpha \in R$
\begin{align} 
\label{def-RV}
 \lim_{x \to \infty}\frac{f(tx)}{f(x)}=t^\alpha , \quad  \forall t>0. 
\end{align}
We sometimes use the notation $ f \in \mbox{RV}_\alpha$.
The number $ \alpha $ in the above is called the index of regular
variation. A function satisfying \eqref{def-RV} with $\alpha=0$ is
called slowly varying.

Let $ \beta: R^+ \to R^+ $ be positive, continuously
differentiable, monotone decreasing, integrable (i.e.\ $ \int_0^\infty
\beta(r)dr<\infty$) and regular varying with index $-1$.  A typical $
\beta(\eta)$ is
\begin{align} 
\label{beta1}
\frac{1}{\eta+c}\frac{1}{\{\mathrm{Log}_n(\eta+c)\}^2}
\prod_{i=0}^{n-1}\frac{1}{\mathrm{Log}_i(\eta+c)},
\end{align}
where $n$ is a positive integer, $\mathrm{Log}_0(\eta+c) \equiv 1 $,
\begin{align*}
\mathrm{Log}_i(\eta+c)=\underbrace{\log \log \cdots \log}_{i}(\eta+c),
\quad i \ge 1,
\end{align*}
and $c$ is chosen such that $  \mathrm{Log}_n(c)>0 $.
Note that for
\eqref{beta1}
\[
 \int_\eta^\infty \beta(r)dr= \frac{1}{\mbox{Log}_n(\eta+c)}.
\]

The following results for $\beta(\eta)$ satisfying the above
assumptions are known from the theory of regularly varying functions.
% See Geluk and de Haan for the detail on regularly (slowly) varying
% functions. 
\begin{lemma} \label{karamata}
\begin{enumerate}
 \item $ \int_{\eta}^\infty \beta(r)dr \in \mbox{\rm RV}_0$ and 
$ \beta'(r) \in \mbox{\rm RV}_{-2}$.
\item $ \lim_{\eta \to \infty} \eta \beta(\eta)/\int_{\eta}^\infty
       \beta(r)dr=0$, 
$ \lim_{\eta \to \infty} \eta \beta'(\eta)/\beta(\eta)=-1$.
%and \\
%$ \lim_{\eta \to \infty} \eta \beta''(\eta)/\beta'(\eta)=-2$.
\end{enumerate} 
\end{lemma}

We now define  functions $H_i(\eta)$, $i=1,2,\ldots$, based on $
\beta(r)$ by 
\begin{align} 
\label{H}
H_i(\eta)=
\frac{\int_\eta^\infty e^{ (\eta-r)/i}\beta(r)dr} 
{\int_\eta^\infty \beta(r)dr}.
\end{align}
These functions are very useful for constructing a sequence of proper
prior densities approaching the target improper density in the next
section.  The properties of $H_i$ are given in the following theorem.

\begin{thm} 
\label{thm-beta-H}
 \begin{enumerate}
\item  $0 \le H_1(\eta) \le H_2(\eta) \le \cdots \le 1$.  For 
any fixed $ \eta$, 
$ \lim_{i \to \infty}H_i(\eta)=1 $.
\item For any fixed $i$, $ \lim_{\eta \to \infty}
\int_\eta^\infty \beta(r)dr \beta(\eta)^{-1}
H_i(\eta)=i$ and hence $ H_i(\eta) \in \mbox{\rm RV}_{-1}$.
\item For any fixed $\eta$, $ \lim_{i \to
	\infty}H'_i(\eta)=0 $.
\item $  | H'_i(\eta) |  < 2 \beta(\eta)/\int_\eta^\infty
\beta(r)dr$ for all $\eta > 0$.
\item For any $ \epsilon>0$, there exists $ \eta_0$ such that
$ -1 -\epsilon < \eta H'_i(\eta)/H_i(\eta) \leq 0 $ for 
all $\eta \geq \eta_0$ and for all $i$.
\end{enumerate}
\end{thm}

\begin{proof}
It is obvious that $0 \le H_1(\eta)  \le 1$  and $H_i(\eta)$ is
increasing in $i$. For fixed  $  \eta $, 
$H_i(\eta) \uparrow 1$ by the monotone convergence theorem. 

By integration by parts, the numerator of $H_i(\eta)$ is written as
\begin{equation}
\label{eq:numerator-integ-parts}
\int_\eta^\infty e^{ (\eta-r)/i}\beta(r)dr = 
i \beta(\eta) + i \int_\eta^\infty e^{ (\eta-r)/i}\beta'(r)dr.
\end{equation}
Therefore 
\begin{align}
 H_i(\eta)=i \frac{\beta(\eta)}{\int_\eta^\infty \beta(r)dr} 
+i
\frac{\int_\eta^\infty e^{ (\eta-r)/i}\beta'(r)dr}
{\int_\eta^\infty \beta(r)dr}.
\label{H+}
\end{align}
\eqref{H+} divided by \eqref{H} is
\begin{align*}
 1 = 
i \frac{\beta(\eta)}{H_i(\eta)\int_\eta^\infty \beta(r)dr} 
+i
\frac{\int_\eta^\infty e^{ -r/i}\beta'(r)dr}
{\int_\eta^\infty e^{ -r/i}\beta(r)dr} .
\end{align*}
For fixed $i$, the second term of the above equation converges to $0$
as $ \eta \rightarrow \infty $ by the L'Hospital theorem.  $H_i(\eta)
\in \mbox{RV}_{-1}$ because $ \int_\eta^\infty \beta(r)dr \in
\mbox{RV}_0$ and $ \beta(\eta) \in \mbox{RV}_{-1}$.


Using (\ref{eq:numerator-integ-parts}) again, differentiation of
the numerator of $H_i(\eta)$ gives
\[
\left( \int_\eta^\infty e^{ (\eta-r)/i}\beta(r)dr\right)'=
\frac{1}{i}\int_\eta^\infty e^{ (\eta-r)/i}\beta(r)dr- \beta(\eta)
= \int_\eta^\infty e^{ (\eta-r)/i}\beta'(r)dr .
\]
Therefore
\begin{align}
 H'_i(\eta) 
=
\frac{\beta(\eta)\int_\eta^\infty e^{ (\eta-r)/i}\beta(r)dr}
{(\int_\eta^\infty \beta(r)dr)^2}
-\frac{\int_\eta^\infty e^{ (\eta-r)/i}\{-\beta'(r)\}dr}
{\int_\eta^\infty \beta(r)dr}.
 \label{bibun-H}  
\end{align}
Note that $-\beta'(r)\ge 0$ by our assumption.  Each term of the right
hand side of \eqref{bibun-H} is nondecreasing in $i$ and hence by the
monotone convergence theorem
\begin{align*}
 \lim_{i \to \infty}H'_i(\eta)
= 
\frac{\beta(\eta)\int_\eta^\infty \beta(r)dr}
{(\int_\eta^\infty \beta(r)dr)^2}
-\frac{\int_\eta^\infty \{-\beta'(r)\}dr}
{\int_\eta^\infty \beta(r)dr}
 =0 .
\end{align*}
Furthermore we have
\begin{eqnarray*}
 \left| H'_i(\eta)\right| 
& < &
\left|
\frac{\beta(\eta)\int_\eta^\infty e^{ (\eta-r)/i}\beta(r)dr}
{(\int_\eta^\infty \beta(r)dr)^2}\right|
+\left|\frac{\int_\eta^\infty e^{ (\eta-r)/i}\{-\beta'(r)\}dr}
{\int_\eta^\infty \beta(r)dr}\right|\\
& <& 2\beta(\eta)/\int_\eta^\infty \beta(r)dr.
\end{eqnarray*}

Dividing \eqref{bibun-H} by \eqref{H}, we have
\begin{align}
 \eta \frac{H'_i(\eta)}{H_i(\eta)}
&=\eta \left(\frac{\beta(\eta)}{\int_\eta^\infty
\beta(r)dr}+\frac{\int_\eta^\infty e^{ -r/i}\beta'(r)dr}
{\int_\eta^\infty e^{-r/i}\beta(r)dr}\right) \label{bibun-HH} \\
&>  \frac{\eta\beta(\eta)}{\int_\eta^\infty
\beta(r)dr}- \frac{\int_\eta^\infty e^{ -r/i}\{-r \beta'(r)/\beta(r)\}
\beta(r)dr}
{\int_\eta^\infty e^{-r/i}\beta(r)dr} \nonumber \\ 
&>  \frac{\eta\beta(\eta)}{\int_\eta^\infty
\beta(r)dr}- \sup_{r>\eta}\frac{-r \beta'(r)}{\beta(r)}. \nonumber
\end{align}
By 2 of Lemma \ref{karamata} the right hand side converges to $-1$.
This implies that for any $ \epsilon>0$ there exists $ \eta_0$
such that $ \eta H'_i(\eta)/H_i(\eta)>-1-\epsilon$ for all  
$\eta \geq \eta_0$ and for all $i$.
Finally 
%by the covariance inequality (Lemma 6.6, Chapter 5 of
%\cite{Lehmann-Casella-TPE}) 
we will prove that $H'_i(\eta) \leq
0$ for sufficiently large $\eta$ independent of $i$. 
 By 2 of Lemma \ref{karamata}, 
\begin{align*}
 \frac{\eta \int_\eta^\infty \beta(r)dr}{\beta(\eta)}
\left(\frac{\beta(\eta)}{\int_\eta^\infty \beta(r)dr}\right)'
=\eta \frac{\beta'(\eta)}{\beta(\eta)}+ \eta \frac{\beta(\eta)} 
{ \int_\eta^\infty  \beta(r)dr} \to -1
\end{align*}
and hence
$ \beta(\eta)/\int_\eta^\infty
\beta(r)dr$ is eventually nonincreasing.
%\[
%\eta^2 \left( \frac{\beta'(\eta)}{\beta(\eta)}\right)'
%=\eta \frac{\beta''(\eta)}{\beta'(\eta)} \; \eta 
%  \frac{\beta'(\eta)}{\beta(\eta)}
% - \left(\eta \frac{\beta'(\eta)}{\beta(\eta)}\right)^2 \rightarrow
%(-2)(-1) -(-1)^2 = 1.
%\]
%This shows that, $\beta'(\eta)/\beta(\eta)$ is eventually monotone
%nondecreasing.  
Hence by redefining $\eta_0$ if necessary, we can assume
that $ \beta(\eta)/\int_\eta^\infty
 \beta(r)dr$ is monotone nonincreasing for $\eta
\ge \eta_0$. 
By integration by parts on the numerators of each term in
\eqref{bibun-H}, we have
\begin{align*}
& \int_\eta^\infty e^{ -r/i}\beta(r)dr= e^{-\eta/i}\int_\eta^\infty 
\beta(r)dr -i^{-1} \int_\eta^\infty e^{ -r/i} \left\{\int_r^\infty 
\beta(s)ds\right\}dr \\
& \int_\eta^\infty e^{ -r/i}\beta'(r)dr= -e^{-\eta/i}\beta(\eta)
+i^{-1} \int_\eta^\infty e^{ -r/i} \beta(r)dr 
\end{align*}
and hence
\begin{align*}
\left\{  \frac{i \int_\eta^\infty  \beta(r)dr }{\int_\eta^\infty e^{
 -r/i} \left\{\int_r^\infty  
\beta(s)ds\right\}dr} \right\}
H'_i(\eta)
= -\frac{\beta(\eta)}{\int_\eta^\infty \beta(r)dr}+
\frac{\int_\eta^\infty e^{ -r/i} \beta(r)dr}{\int_\eta^\infty e^{
 -r/i} \left\{\int_r^\infty  \beta(s)ds\right\}dr},
\end{align*}
which is nonpositive for $ \eta \ge \eta_0$.
%\begin{eqnarray*}
%&&
%\int_\eta^\infty e^{(\eta-r)/i}
%\left(-\frac{\beta'(r)}{\beta(r)}\right) 
%\frac{\beta(r)}{\int_\eta^\infty \beta(r')dr'}  dr
%\\ && \qquad
%\ge
%\int_\eta^\infty e^{(\eta-r)/i}
%\frac{\beta(r)}{\int_\eta^\infty \beta(r')dr'} dr \times 
%\int_\eta^\infty 
%\left(-\frac{\beta'(r)}{\beta(r)}\right) 
%\frac{\beta(r)}{\int_\eta^\infty \beta(r')dr'} dr. 
%\end{eqnarray*}
%Comparing this to the first equality in (\ref{bibun-HH}), we see that
%$H_i'(\eta)/H_i(\eta) \le 0$ for $\eta \ge \eta_0$ and for all $i$.
%
%By Lemma \ref{karamata}, 
%By \eqref{bibun-HH}, 
\end{proof}

\subsection{Asymptotic behavior of expectations}
\label{sec:asymptotic-behavior}
In the next section, we need evaluation of an asymptotic
behavior of expectation
\begin{align*}
 E_x[\rho(\theta)]=\int_{R^p} \rho(\theta)f(\|\theta - x \|)d\theta
\end{align*}
for sufficiently large $\|x\|$, where a random vector $ \theta$ has
the density function $ f(\|\theta - x\|)$.  This is the expected value
with respect to the posterior distribution. Interchanging the roles of
$x$ and $\theta$, in this subsection, 
we consider the asymptotic behavior of expectation
\begin{align*}
 E_\theta[\rho(X)]=\int_{R^p} \rho(x)f(\| x - \theta \|)dx
\end{align*}
for sufficiently large $\|\theta\|$, where a random vector $ X$ has the
density function $ f(\|x - \theta \|)$.  We believe that this does not
confuse the readers.  

We discuss some notations used in the following. In addition to the
Euclidean norm $\|x\|^2=x_1^2 + \cdots + x_p^2$, we consider the norm
$\|x\|_d^2=d_1^2 x_1^2 + \cdots + d_p^2 x_p^2$.  For convenience we assume,
without loss of generality, that $ d_1 \ge \cdots \ge d_p \ge 1.  $ Under
this assumption
\begin{equation}
\label{eq:norm-d}
 \| x \| \le \| x \|_d \le d_1 \| x \|.
\end{equation}
By introducing this norm our results hold for
elliptically-contoured distributions.  
The gradient of $\rho(x)$ is denoted by
\[
\nabla \rho(x)=(\frac{\partial}{\partial x_1} \rho(x),\ldots,\frac{\partial}{\partial
x_p}\rho(x))' .
\]
We also write $\nabla_j \rho(x)=(\partial/\partial x_j) \rho(x)$.  Finally 
we write $c_p=2\pi^{p/2}/\Gamma(p/2)$.

Now we make the following regularity conditions on the density $f$ and
the function $\rho $. 
\begin{description}
\item[F1] There exist $r_0>0$, $L>0$, and $s>1$, such that
  $r^{p+s}f(r) \leq L$ for all $r\geq r_0$.
% \item[F2] There exists $r_1$ such that  $f(r)$ is monotone
% 	   nonincreasing for all $r\geq r_1$.
\item[B1] 
% $\rho(x)$ is elliptically symmetric with respect to the
% origin, that is, 
  $\rho(x)$ is written as $ \rho(x)=\varrho(\| x \|_d)$, where 
$\varrho(r)$ is continuously differentiable in $r>0$.
\item[B2]  There exists $ r_1 \ge 1$ and $ t_1 \leq t_2$
such that
$ \varrho(r)>0 $ and $ t_1 \leq r\varrho'(r)/\varrho(r) \leq t_2$ for
	   all $r \geq r_1$.
%\item[B3] $ \int_0^1 w^{p+a-1}| \varrho(r)|dr <\infty$.  ($a=0$?)
%$\int_{\|x\|_d<1 }\|x\|_d^a |\rho(x)|dx<\infty$ for $a=0$ or $1$.
%\item[B3] $ \varrho(r)$ is continuously differentiable.
\end{description}
Assumption {\bfseries B2} is, for instance, satisfied by 
\begin{align*}
 \varrho(r)=\exp\left( \int_0^r \frac{-p+\alpha\cos t}{t+1} dt \right), 
\mbox{ for }\alpha>0
\end{align*}
where we easily see 
$ r\varrho'(r)/\varrho(r)=\{r/(r+1)\}(-p+\alpha \cos r)$ and hence
$t_1=-p-\alpha$ and $t_2=-p+\alpha$ in {\bfseries B2}.
Since $ \lim_{r \to \infty} r\varrho'(r)/\varrho(r)$ exists for 
regularly varying $\varrho$, 
we deal with a broader class of functions
than the class of regularly varying functions.
We will discuss more in Section 3.
Note that
\[
\nabla \rho(x)=\frac{\varrho'(\| x \|_d)}{\| x \|_d} 
(d_1^2 x_1, \ldots, d_p^2 x_p)'
\]
and
\[
\| \nabla \rho(x)\| = \frac{|\varrho'(\| x \|_d)|}{\| x \|_d} (d_1^4 x_1^2 +
\cdots + d_p^4 x_p^2)^{1/2}\le d_1 |\varrho'(\| x \|_d)|.
\]

The following lemma is useful. The proof based on the integration of
$(\log \varrho(r))'=\varrho'(r)/\varrho(r)$ is easy and omitted.
\begin{lemma}
\label{lem:b3}
Under the assumption {\bfseries B2}
\begin{align*}
 (z/y)^{t_1}\leq \varrho(z)/\varrho(y) \leq (z/y)^{t_2}
\end{align*}
for any $ z>y \geq r_1$. Moreover 
\[
\limsup_{y \to \infty}\sup_{\alpha y \leq z \leq \beta y} 
\varrho(z)/\varrho(y) \le 
\max(\alpha^{t_1},\beta^{t_2})\]
for any $ 0<\alpha<1<\beta $.
\end{lemma}


We now state the following theorem concerning the asymptotic behavior
of $E_\theta[\rho(X)]$ for large $\|\theta\|_d$.

\begin{thm} 
\label{thm-zenkin}
Assume {\bf F1}, {\bf B1} and {\bf B2}.  
For $a=0$ or $1$, and $j=1,\dots,p$, 
if $ s> \max(1, -t_1-a-p, t_2+a)$ and
$ \int_0^1 r^{p+a-1}| \varrho(r)|dr <\infty$,
then there exists $ \epsilon>0$ (say 
$\epsilon=  \min(1, s+t_1+a+p)/4$)
 such that
\begin{equation}
\label{eq:thm22}
\|\theta\|_d^{\epsilon-a}
\left|E_\theta[X_j^a \rho(X)] - \theta_j^a \rho(\theta)
 \right| 
% = \|\theta\|_d^{\epsilon-a}
% \left|E_\theta\left[X_j^a \left(\frac{\rho(X)}{\rho(\theta)} - 1
% \right)\right] \right|
< C \rho(\theta)
\end{equation}
for  $\|\theta\|_d \geq 2d_1\max(r_0,r_1)$.
Moreover $C$ depends on $ \rho$ (or $ \varrho$) only through
$r_1$, $t_1$, $t_2$ and
$ \{\varrho(r_1)\}^{-1}\int_0^{r_1}r^{p+a-1}|\varrho(r)|dr$.
\end{thm}
For simplicity, in the rest of the paper, we will write 
$ E_\theta[X_j^a \rho(X)] \approx \theta_j^a \rho(\theta)$
if, as in Theorem \ref{thm-zenkin}, there exists $\epsilon>0$ such that
\eqref{eq:thm22} is satisfied for sufficiently large $\|\theta\|_d$.
\begin{proof}
%[The proof of Theorem \ref{thm-zenkin}]
Note that 
\[E_\theta[X_j^a \rho(X)] - \theta_j^a \rho(\theta)=
E_\theta[X_j^a (\rho(X)-\rho(\theta))].
\]
 Fix $0 < \nu < 1$ (set $\nu=1/2$ finally).
Define
\begin{eqnarray*}
V_{\nu}&=&\{ x: \|x-\theta \| \leq \nu\| \theta \| \} \\
V'_{\nu}&=&\{ x: (1-\nu)\|\theta\| \leq \|x\| \leq (1+\nu) \|\theta \| \}\\
V''_{\nu}&=&\{ x: d_1^{-1}(1-\nu)\|\theta\|_d \leq \|x\|_d \leq
d_1 (1+\nu) \|\theta \|_d \} .
\end{eqnarray*}
By (\ref{eq:norm-d}) $ V_\nu \subset V'_\nu \subset V''_\nu$.
Then 
\begin{align}
& 
\|\theta\|_d^{\epsilon-a}
\left|E[X_j^a (\rho(X)-\rho(\theta))] \right| \nonumber \\
&  \leq \|\theta\|_d^{\epsilon-a}
\left( \int_{V_\nu} + \int_{V_\nu^C} \right) \|x\|^a 
\, |\rho(x)-\rho(\theta) | \,
f(\|x -\theta\|)dx 
%+
%\|\theta\|_d^{\epsilon-a}
%\int_{V_\nu^C} \|x\|^a \,|\rho(x)- \rho(\theta)|\,
%f(\|x -\theta\|)dx 
\nonumber \\ 
&  
\leq \|\theta\|_d^{\epsilon}
\sup_{x \in V''_\nu}
\left(\frac{\|x\|_d}{\|\theta\|_d}\right)^a
\int_{V_\nu} | \rho(x) - \rho(\theta)| \,
f(\|x-\theta\|)dx
\nonumber \\
&\quad + \|\theta\|_d^{\epsilon-a} \rho(\theta) \int_{V_\nu^C}
\|x\|^af(\|x -\theta\|)dx 
%\nonumber \\ &  \qquad\quad
+ \|\theta\|_d^{\epsilon-a} 
\int_{V_\nu^C} \|x\|^a | \rho(x)|  
f(\|x -\theta\|)dx 
\nonumber \\ 
& =  I_1 + I_2 + I_3.  \quad ({\rm say})
\label{eq:intergral-3-parts}
\end{align}
%\begin{align}
%& 
%\|\theta\|_d^{\epsilon-a}
%\left|E[X_j^a (\rho(X)-\rho(\theta))] \right| \nonumber \\
%& \qquad \leq \|\theta\|_d^{\epsilon-a}
%\int_{V_\nu} \|x\|_d^a \, |\rho(x)-\rho(\theta) | \,
%f(\|x -\theta\|)dx 
%\nonumber \\ & \qquad\quad+
%\|\theta\|_d^{\epsilon-a}
%\int_{V_\nu^C} \|x\|^a \,|\rho(x)- \rho(\theta)|\,
%f(\|x -\theta\|)dx 
%\nonumber \\ & \qquad 
%\leq \|\theta\|_d^{\epsilon}
%\sup_{x \in V''_\nu}
%\left(\frac{\|x\|_d}{\|\theta\|_d}\right)^a
%\int_{V_\nu} | \rho(x) - \rho(\theta)| \,
%f(\|x-\theta\|)dx
%\nonumber \\
%&  \qquad\quad + \|\theta\|_d^{\epsilon-a} \rho(\theta) \int_{V_\nu^C}
%\|x\|^af(\|x -\theta\|)dx 
%\nonumber \\ &  \qquad\quad
%+ \|\theta\|_d^{\epsilon-a} 
%\int_{V_\nu^C} \|x\|^a | \rho(x)|  
%f(\|x -\theta\|)dx 
%\nonumber \\ 
%&\qquad =  I_1 + I_2 + I_3.  \quad ({\rm say})
%\label{eq:intergral-3-parts}
%\end{align}

Consider the first integral $I_1$.  
$ \sup_{x \in V''_\nu}
\left(\|x\|_d/\|\theta\|_d \right)^a  \leq d_1^{a}(1+\nu)^{a}$.
If  $s>1$, then
$m_1=\int_{R^p} \|x-\theta\|f(\|x - \theta\|)dx$ is finite.
Therefore for $ \|\theta\|_d \geq d_1(1-\nu)^{-1}r_1$  we have
\begin{align*}
  & \|\theta\|_d^{\epsilon} \int_{V_\nu} \,|
  \rho(x)-\rho(\theta)| \,
  f(\|x - \theta\|)dx \\
  &\qquad = \|\theta\|_d^{\epsilon} \int_{V_\nu} |(x-\theta)'\nabla
  \rho(x^*)|f(\|x - \theta\|)dx , \
  x^* \in V_\nu \\
  &\qquad \leq m_1 \|\theta\|_d^{\epsilon}
  \sup_{x \in V_\nu} \| \nabla \rho(x) \| \\
  &\qquad \leq m_1 d_1 \|\theta\|_d^{\epsilon} \sup_{x \in
    V_\nu}
  |\varrho'(\|x\|_d)| \\
  &\qquad \leq m_1d_1 \|\theta\|_d^{\epsilon-1} \sup_{x \in
    V''_\nu}\frac{\| \theta \|_d}{\|x\|_d} \sup_{x \in
    V''_\nu}\frac{\varrho(\|x\|_d)}{\varrho(\|\theta\|_d)} \sup_{x \in
    V''_\nu} \frac{\|x\|_d | \varrho'(\|x\|_d) |}{\varrho(\|x\|_d)}
  \times \rho(\theta)
  \\
%  &\qquad = \max(|t_1|, |t_2|) m_1 \frac{d_1}{1-\nu}
%\max\left( \left(\frac{d_1}{1-\nu}\right)^{t_1}, 
%\left(d_1(1+\nu)\right)^{t_2} \right)
% \| \theta  \|_d^{\epsilon -1}\times \rho(\theta) \\
  & \qquad \le 
%t_1 m_1 d_1^{2+t_1} (1-\nu)^{-t_1-1} 
  \frac{m_1 d_1^2}{1-\nu}
 \max\left( \left(\frac{1-\nu}{d_1}\right)^{t_1}, 
\left\{d_1(1+\nu)\right\}^{t_2} \right) 
\max(|t_1|, |t_2|)
\times \rho(\theta)
%  \qquad ()
\end{align*}
for $ 0<\epsilon<1$.
Therefore we have $I_1 \le C_{1}\rho(\theta)$, where
$C_{1}= d_1^a(1+\nu)^{a} 
 m_1 \{d_1^2/(1-\nu)\}
\max( \{(1-\nu)/d_1\}^{t_1},
\{d_1(1+\nu)\}^{t_2}) \max(|t_1|, |t_2|)$.

Now we consider the integral outside of $V_\nu$.  We only consider 
$\|\theta\|_d \ge \max(d_1 \nu^{-1} r_0,r_1)$.  
%
%
Then  for $x\in V_\nu^C$
\[
\| x- \theta \| \ge \nu \| \theta\| \ge \nu d_1^{-1} \| \theta\|_d \ge
r_0.
\]
Therefore we have, for $0\le \alpha < s$
\begin{align}
  \int_{V_\nu^C}\|x\|^\alpha
  f(\|x - \theta \|)dx 
&\leq  \int_{V_\nu^C}\{\|x-\theta \|+\|\theta\|\}^\alpha
  f(\|x - \theta \|)dx \notag \\
& \leq (1+1/\nu)^{\alpha}\int_{V_\nu^C}\|x-\theta \|^\alpha
  f(\|x - \theta \|)dx \notag \\
& \leq 
(1+1/\nu)^{\alpha}c_p L  \int_{\nu \|\theta\|}^\infty
r^{-s+\alpha-1} dr \notag \\
%\{  + \|\theta\|_d^\alpha \, r^{-s-1} \}dr \notag \\
&=
(1+1/\nu)^{\alpha}c_p L  \frac{(\nu \|\theta\|)^{-s+\alpha}}{s-\alpha}
\notag\\
&\le 
C_2(\alpha)\|\theta\|_d^{\alpha- s},
%(1+1/\nu)^{\alpha}\left(\frac{d_1}{\nu}\right)^{s-\alpha}\frac{ c_p L}{s-\alpha}
% \frac{d_1^{s}}{\nu^s}
%\left(\frac{\nu^{\alpha}d_1^{-\alpha}}{s-\alpha}+
%\frac{1}{s} \right)
\label{soto-sekibun}
\end{align}
where $ C_2(\alpha)=(1+1/\nu)^{\alpha}c_p L (d_1/\nu)^{s-\alpha}
 (s-\alpha)^{-1}$.
%Therefore for the second term $I_2$, omitting
%$\rho(\theta)$, we have
%\begin{align*}
%& \| \theta \|_d^{\epsilon-a} \int_{V_\nu^C}\|x\|^a
%  f(\|x - \theta \|)dx \\
%&\leq \| \theta \|_d^{\epsilon-a} \int_{V_\nu^C}(\|x-\theta \|^a+\|\theta\|^a)
%  f(\|x - \theta \|)dx \\
%& \leq 
%c_p L \|\theta\|_d^{\epsilon-a} \int_{\nu \|\theta\|}^\infty
%\{ r^{-s+a-1} + \|\theta\|_d^a \, r^{-s-1} \}dr \\
%&=
%c_p L \|\theta\|_d^{\epsilon-a} \left(\frac{(\nu \|\theta\|)^{-s+a}}{s-a}+
%   \|\theta\|_d^a   \frac{(\nu\|\theta\|)^{-s}}{s} \right)\\
%&\le  c_p L \frac{d_1^{s}}{\nu^s}
%\left(\frac{\nu^{a}d_1^{-a}}{s-a}+
%\frac{1}{s} \right)\|\theta\|_d^{\epsilon- s} .
%\end{align*}
Hence for the second term $I_2$, 
if $s>1$ and $0<\epsilon<1$, 
then  $I_2 \le C_{2}(a) \rho(\theta)$
%, where
%$C_{2}=(1+1/\nu)^{\alpha}c_p L (d_1/\nu)^{s-a} (s-a)^{-1}$
 for $a=0,1$.

We have seen that $I_1$ and $I_2$ are bounded from above assuming only $s>1$.

The third term $I_3$  of (\ref{eq:intergral-3-parts}) is more
problematic.  
Write 
\begin{eqnarray*}
I_3 &=&
\|\theta\|_d^{\epsilon-a}  \int_{V_\nu^C}
\|x\|^a|\rho(x)|f(\|x -\theta\|)dx \\
&\le &  \|\theta\|_d^{\epsilon-a}
 \left( 
 \int_{V_\nu^C \cap \{\|x\|_d < r_1 \}} 
 + \int_{V_\nu^C \cap  \{r_1 \leq \|x\|_d \leq \|\theta\|_d \}}  \right.
\\ 
  && \qquad \qquad  \qquad\qquad 
  \left. 
  +\int_{V_\nu^C \cap \{\|x\|_d > \|\theta \|_d \} } \right) \|x\|^a
  |\rho(x)|  f(\|x -\theta\|)dx \\
&=& I_{31} + I_{32} + I_{33}. \qquad ({\rm say})
\end{eqnarray*}
We take care of $I_{33}$ first.  
Since $\varrho(r)r^{-t_2}$ is monotone nonincreasing for $ r \ge r_1$, 
$\rho(x)\|x\|_d^{-t_2} \le \rho(\theta)\| \theta \|_d^{-t_2}$ 
for $\|x\|_d > \|\theta \|_d(\ge r_1)$.
Therefore 
%using \eqref{soto-sekibun} 
we have, 
%for $s>\max(1,a+t_2)$ and $0<\epsilon<1$,
\begin{align*}
I_{33} & \le 
\|\theta\|_d^{\epsilon-a-t_2}  \rho(\theta) d_1^{\max(t_2,0)}
\int_{V_\nu^C \cap \{\|x\|_d > \|\theta \|_d \} } \|x\|^{a+t_2} 
f(\|x -\theta\|)dx .
\end{align*}
If $ 0 \leq a+t_2<s$, as in \eqref{soto-sekibun}
\begin{align*}
\int_{V_\nu^C \cap \{\|x\|_d > \|\theta \|_d \} } \|x\|^{a+t_2} 
f(\|x -\theta\|)dx &\leq \int_{V_\nu^C} \|x\|^{a+t_2} 
f(\|x -\theta\|)dx \\ & \leq C_2(a+t_2) \|\theta \|_d^{a+t_2-s}
 \end{align*} 
and if $ a+t_2<0$,
\begin{align*}
\int_{V_\nu^C \cap \{\|x\|_d > \|\theta \|_d \} } \|x\|^{a+t_2} 
f(\|x -\theta\|)dx &\leq d_1^{-t_2-a} \|\theta\|^{a+t_2}_d 
\int_{V_\nu^C} 
f(\|x -\theta\|)dx \\ & \leq d_1^{-t_2-a} C_2(0) \|\theta \|_d^{a+t_2-s}.
 \end{align*}
%\begin{align*}
%&\int_{V_\nu^C \cap \{\|x\|_d > \|\theta \|_d \} } \|x\|^{a+t_2} 
%f(\|x -\theta\|)dx \\
%&
%\begin{cases}
% \leq \int_{V_\nu^C} \|x\|^{a+t_2} 
%f(\|x -\theta\|)dx  & \\ 
%\leq C_2(a+t_2) \|\theta \|^{a+t_2-s}
% & \mbox{ if }0 \leq a+t_2<s \\
%\leq d_1^{-t_2-a} \|\theta\|^{a+t_2}_d 
%\int_{V_\nu^C} 
%f(\|x -\theta\|)dx & \\ 
%\leq C_2(0) \|\theta \|^{a+t_2-s}
% & \mbox{ if }  a+t_2<0 
%\end{cases}
%\end{align*}
%
%& \le \|\theta\|_d^{\epsilon-s} C_{33} \rho(\theta) 
% \le C_{33} \rho(\theta) 
%\|\theta\|_d^{\epsilon-s}  
%d_1^{t_2}
%c_p L (d_1/\nu)^s \{(\nu/d_1)^{a+t_2}(s-a-t_2)^{-1}+1/s
%\}
%\times \rho(\theta) \\
%& \le 
%C_{33} 
%\times \rho(\theta) 
Hence $I_{33}\leq C_{33}\rho(\theta)$ where $ C_{33}= d_1^{\max(t_2,0)}
\max( C_2(a+t_2), C_2(0)d_1^{-a-t_2})$.

Next we consider $I_{31}$.
For $\|\theta\|_d \ge \max(d_1 \nu^{-1}
r_0,r_1)$ and $x\in V_\nu^C$
\begin{align}
f(\| x- \theta \|) \le  L \| x- \theta \|^{-p-s} 
\le L(\nu \|\theta\|)^{-p-s} 
\leq L(d_1/\nu\| \theta\|_d)^{p+s}. \label{f-inequality}
%\nu^{-p-s}^{-p-s}.
\end{align}
Therefore
\[
I_{31}\le 
\|\theta\|_d^{\epsilon-a} 
Ld_1^{p+s}\nu^{-p-s}\| \theta\|_d^{-p-s}
\int_{\|x\|_d \leq r_1}  \|x\|_d^a \, \rho(x) dx.
\]
Note that by simple change of variables we have
\[
\frac{\partial}{\partial r} \int_{ \|x\|_d \le r} dx = c_{p,d} r^{p-1},
\qquad c_{p,d}=c_p \prod_{i=1}^p d_i^{-1} .
\]
Then
\[
\int_{\|x\|_d \leq r_1}  \|x\|_d^a \, |\rho(x)| dx 
= c_{p,d} \int_0^{r_1} r^{p+a-1}
|\varrho(r)| dr.
\]
Therefore 
\[
I_{31} \le C_* \|\theta\|_d^{\epsilon-a-p-s}  
 \int_0^{r_1} r^{p+a-1}
|\varrho(r)| dr,
\]
where $C_*=Ld_1^{p+s}\nu^{-p-s}
c_{p,d}$.
On the other hand for  $\|\theta\|_d \ge  r_1$, 
$\rho(\theta)=\varrho(\|\theta\|_d)$ is bounded from below as
\begin{align*}
\varrho(r_1)r_1^{-t_1}\|\theta\|_d^{t_1} \leq  \varrho(\|\theta\|_d).
%\leq \varrho(r_1)r_1^{t_2}\|\theta\|_d^{-t_2}.
\end{align*}
Therefore
\begin{align*}
I_{31} \le  \|\theta\|_d^{\epsilon-a-p-s - t_1} \times
C_*
%Lc_{p,d} \frac{d_1^{p+s}}{\nu^{p+s}} \times
\frac{r_1^{t_1}}{\varrho(r_1)} 
 \int_0^{r_1} r^{p+a-1}
|\varrho(r)| dr
\times \rho(\theta).
\end{align*}
%where 
%\[
%C_{31} =  .
%\]
Hence if $s> -t_1 - a - p$, then we can choose $\epsilon >0$ 
(say $ \epsilon=(a+p+s+t_1)/4$) such
that $ \epsilon-a-p-s-t_1<0$ and hence $ I_{31} \le C_{31}\rho(\theta)$ 
where
\[
 C_{31} =
%Lc_{p,d} \frac{d_1^{p+s}}{\nu^{p+s}} \times
C_*
 \frac{r_1^{t_1}}{\varrho(r_1)}\int_0^{r_1} r^{p+a-1}
|\varrho(r)| dr.
\]

Finally we consider $I_{32}$.  
Note $\varrho(r)\le \varrho(\|\theta\|_d)\|\theta\|_d^{-t_1} r^{t_1}$ 
for $r_1 \le r \le \|\theta\|_d$ and
\eqref{f-inequality}.
Then
\begin{align*}
I_{32} 
%&\le
%\|\theta\|_d^{\epsilon-a}
%\int_{r_1 \leq \|x\|_d \leq \|\theta\|_d}  \|x\|_d^a \, \rho(x) \,
%f(\| x- \theta \|)dx
%\\
&\leq 
\|\theta\|_d^{\epsilon-a-t_1}
Ld_1^{p+s}\nu^{-p-s}\| \theta\|_d^{-p-s}
\varrho(\|\theta\|_d)
\int_{r_1 \leq \|x\|_d \leq \|\theta\|_d}  \|x\|_d^{t_1+a} dx
\\
%&=
%\|\theta\|_d^{\epsilon-a-p-s} \times
%%Ld_1^{p+s}\nu^{-p-s} 
%C_*
%\varrho(r_1)r_1^{-t_2}  \times
%\int_{r_1}^{\|\theta\|_d}  r^{p+a+t_2-1}dr\\
&\leq \|\theta\|_d^{\epsilon-a-p-s-t_1} \times
C_*
\int_{r_1}^{\|\theta\|_d}  r^{p+a+t_1-1}dr \times \rho(\theta).
\end{align*}
Consider the integral $Q=\int_{r_1}^{\|\theta\|_d} r^{p+a+t_1-1}dr$.  If
$p+a+t_1 < 0$, then 
\[
Q \le r_1^{t_1 + a +p}/(-t_1-a-p).
% \le \frac{-1}{t_2+a+p}.
\]
%\[
%Q \le \frac{-1}{t_2+a+p} r_1^{t_2 + a +p}.
%% \le \frac{-1}{t_2+a+p}.
%\]
Therefore as in the case of $I_{31}$, 
%for some $C_{32}$ 
if $s> -t_1 - a - p$, then we can choose $\epsilon>0$ 
(say $ \epsilon=(a+p+s+t_1)/4$) such
that $ \epsilon-a-p-s-t_1<0$ and hence 
%$ I_{31} \le C_{31}\rho(\theta)$ 
\begin{align*}
 I_{32} \leq r_1^{\epsilon-s} C_{32}
 \rho(\theta) \leq C_{32}
 \rho(\theta)
\end{align*}
where $C_{32}=\{-1/(p+a+t_1)\}C_* $. 
If $ p+a+t_1 \geq 0$, 
\begin{align*}
 Q =\int_{r_1}^{\|\theta\|_d} r^{p+a+t_1+\epsilon_3-1-\epsilon_3}dr
\le
 \frac{\|\theta\|_d^{p+a+t_1+\epsilon_3}r_1^{-\epsilon_3}}
{p+a+t_1+\epsilon_3} 
\end{align*}
for any $ \epsilon_3>0$. 
Hence
\begin{align*}
 I_{32} \leq \frac{C_*}{p+a+t_1+\epsilon_3}
\|\theta\|_d^{\epsilon+\epsilon_3-s}\rho(\theta).
\end{align*}
If $ s>1$, we can choose $\epsilon$ and $ \epsilon_3$ 
(say $ \epsilon=\epsilon_3=1/4$)
such that
$ \epsilon+\epsilon_3-s<0$ and hence
\begin{align*}
 I_{32} \leq C_{32}\rho(\theta),
\end{align*}
where $C_{32}=(p+a+t_1+\epsilon_3)^{-1}C_*$.
%
% If $p+a+t_2 = 0$, then $Q \le \log(\|\theta\|_d)$.  Even in this case,
% if $s > -t_1 - a - p$, we can 
%choose $\epsilon>0$ such that $\|\theta\|_d^{\epsilon-a-p-s - t_1}Q$ is
%bounded and hence we have
%\begin{align*}
% I_{32} 
%\leq r_1^{\epsilon-s} 
%Ld_1^{p+s}\nu^{-p-s} 
% c_{p,d} \rho(\theta).
%\end{align*}
%On the other hand, if $p+a+t_2 >0$, then
%$Q=O(\|\theta\|_d^{p+a+t_2})$ and we need $s>t_2 - t_1$ to choose
%$\epsilon$ such that $I_{32}\le C_{32}\rho(\theta)$.

We have now confirmed that if $s> \max(1, -t_1-a-p,a+t_2)$, 
there exist
$\epsilon>0$ and $C=C_1+C_2+C_{31}+C_{32}+C_{33}$, 
such that (\ref{eq:thm22}) folds for 
$\|\theta\|_d \ge 
\max(d_1 \nu^{-1} r_0,r_1, d_1(1-\nu)^{-1}r_1)$ which equals to
$ 2d_1\max(r_0,r_1)$ for $ \nu=1/2$.
\end{proof}

In the next section, we need asymptotic behavior of the expectation 
of $ \rho(X) \times h_i^\gamma(X)$ where $h_i(\theta)=H_i(\|\theta\|_d)$
given by \eqref{H} and $\gamma>0$.
\begin{corollary} \label{th2.2coro}
Assume {\bf F1}, {\bf B1} and {\bf B2}.  
For $a=0$ or $1$, $i\geq 1$, $\gamma>0$ and $j=1,\dots,p$, 
if $ 
s> \max(1, \gamma-t_1-a-p,t_2+a)$ and
$ \int_0^1 r^{p+a-1}| \varrho(r)|dr <\infty$,
there exists $ \epsilon>0$ (say $ \epsilon=\min(1,s+t_1+a+p-\gamma)
/4$) 
 such that
\begin{equation}
\label{eq:coro22}
\|\theta\|_d^{\epsilon-a}
\left|E_\theta[X_j^a \rho(X)h_i^\gamma(X)] - \theta_j^a 
\rho(\theta)h_i^\gamma(\theta)
 \right| 
% = \|\theta\|_d^{\epsilon-a}
% \left|E_\theta\left[X_j^a \left(\frac{\rho(X)}{\rho(\theta)} - 1
% \right)\right] \right|
< C \rho(\theta)h_i^\gamma(\theta)
\end{equation}
for  $\|\theta\|_d \geq 2d_1\max(r_0, r_1,\eta_0)$. 
%s> \max(1,t_1-a-p)$.
Moreover $C$ does not depend on $i$.
\end{corollary}
\begin{proof}
%[The proof of Corollary \ref{th2.2coro}]
Since 
we have
\[
 \eta\{\varrho(\eta)H_i^\gamma(\eta)\}'/\{\varrho(\eta)H_i^\gamma(\eta)\}
 = \eta 
 \varrho'(\eta)/\varrho(\eta)+\gamma\eta H'_i(\eta)/H_i(\eta),
\]
under Assumption {\bfseries B2} and by 5 of Theorem
 \ref{thm-beta-H},  for $ \epsilon_1= (s+t_1+a+p-\gamma)/16(>0) $
there exists $\eta_0$ such that
\begin{align*}
 t_1-\gamma-\gamma\epsilon_1 \leq 
    \eta \{\varrho(\eta)H_i^\gamma(\eta)\}'/\{\varrho(\eta)H_i^\gamma(\eta)\}
    \leq t_2
\end{align*}
for all $\eta \geq \max(\eta_0, r_1) $. 
Then for $ \epsilon= \min(1,s+t_1+a+p-\gamma)/4(>0) $,
 \eqref{eq:coro22} follows from Theorem \ref{thm-zenkin}.

For  $\lambda_1=\max(\eta_0, r_1) $,
\begin{align*}
 \frac{1}
{H_i^\gamma(\lambda_1)\varrho(\lambda_1)}\int_0^{\lambda_1}r^{p+a-1} 
|H_i^\gamma(r)\varrho(r)|dr \le
\frac{1}{H_1^\gamma(\lambda_1)\varrho(\lambda_1)}\int_0^{\lambda_1}r^{p+a-1} 
|\varrho(r)|dr
\end{align*}
which implies that $C$ does not depend on $i$.
%
%We can therefore  think of
%$ \lambda_1$, $t_1$, $t_2$ replaced by $ir_1$, $t_1-2$, $t_2-2$,
% respectively.
%Let $ \epsilon_2=\min(s+t_1+a+p-2, s-t_2+t_1)/8 $ in part 6 of Theorem
% \ref{thm-beta-H}.
%Then there exists $ \eta_2$ such that $ H_i(i\eta)\geq \nu 
%\eta^{-1-\epsilon_2} i^{-\epsilon_2}$ for all $\eta \geq eta_2$. 
%Define $ \lambda_2$ as $\max(\lambda_1,\eta_2)$.
%We have only to show that 
%\begin{align*}
% \frac{(i\lambda_2)^{\epsilon-a-p-s}}
%{H_i^2(i\lambda_2)\varrho(i\lambda_2)}\int_0^{i\lambda_2}r^{p+a-1} 
%|H_i^2(r)\varrho(r)|dr (=C_{31}(i)),
%\end{align*}
%which corresponds to $C_{31}$ in the proof of Theorem \ref{thm-zenkin}, 
%is bounded from above by a constant independent of $i$ for $i\geq 1$.
%By Lemma \ref{lem:b3}, $ \varrho(i\lambda_2)^{-1}\leq
% i^{-t_1}\varrho(\lambda_2)^{-1}$ and $ \varrho(r) \leq 
%\varrho(\lambda_2)\lambda_2^{-t_2}
%r^{t_2}$ for $r\geq \lambda_2$, we have
%\begin{align*}
%C_{31}(i)
%%&=
%%\frac{(i\lambda_2)^{\epsilon-a-p-s}}{H_i^2(i\lambda_2)\varrho(i\lambda_2)}\int_0^{i\lambda_2}r^{p+a-1}
%%|H_i^2(r)\varrho(r)|dr \\
%& <
%\lambda_2^{\epsilon-a-p-s}
%\frac{i^{\epsilon-a-p-s-t_1}}{H_i^2(i\lambda_2)\varrho(\lambda_2)}\int_0^{i\lambda_2}r^{p+a-1}
%|\varrho(r)|dr \\
%& < \nu^2 \lambda_2^{2+2\epsilon_2+\epsilon-a-p-s}
%\frac{i^{\epsilon+2\epsilon_2-a-p-s-t_1}}
%{\varrho(\lambda_2)}\left\{ \int_0^{\lambda_2}r^{p+a-1}
%|\varrho(r)|dr \right. \\
%& \qquad  \left. + \frac{\varrho(\lambda_2)}{\lambda_2^{t_2}}\int_{\lambda_2}^{i\lambda_2}r^{p+a+t_2-1}
%dr \right\} .
%%& < \nu^2 \lambda_2^{2+2\epsilon_2+\epsilon-a-p-s}
%%\frac{i^{\epsilon+2\epsilon_2-a-p-s-t}}
%%{\varrho(\lambda_2)}\left\{ \int_0^{\lambda_2}r^{p+a-1}
%%|\varrho(r)|dr \right. \\
%%& \qquad  \left. + \int_{\lambda_2}^{i\lambda_2}r^{p+a-1}
%%|\varrho(r)|dr \right\} \\
%\end{align*} 
%If $ p+a+t_2<0$, we have
%$ \int_{\lambda_2}^{i\lambda_2} r^{p+a+t_2-1}dr< -(p+a+t_2)^{-1}
%\lambda_2^{p+a+t_2} $
%and hence 
%$C_{31}(i)< i^{\epsilon+2\epsilon_2-a-p-s-t_1}C'_{31}
% $ for some positive constant $ C'_{31}$.
%Since we have
%\begin{align*}
%&  \epsilon+2\epsilon_2-a-p-s-t_1 \\
%&\leq (s+t_1+a+p-2)/4+ (s+t_1+a+p-2)/4
%-a-p-s-t_1 \\ & < -(a+p+s+t_1)/2 
% <0
%\end{align*}
%we have $C_{31}(i) \leq C'_{31}$ for $i \geq 1$.
%If $ p+a+t_2 \geq 0$, for $ \epsilon_4=(s-t_2+t_1)/4 $
%\begin{align*}
%\int_{\lambda_2}^{i\lambda_2} r^{p+a+t_2+\epsilon_4-1-\epsilon_4}dr
%<
% \lambda_2^{\epsilon_4}(p+a+t_2+\epsilon_4)^{-1}
%(i\lambda_2)^{p+a+t_2+\epsilon_4}
%\end{align*}
%and hence
%\begin{align*}
%C_{31}(i) \leq  i^{\epsilon+2\epsilon_2-a-p-s-t_1}C''_{31}+
%i^{\epsilon+2\epsilon_2-s-t_1+t_2+\epsilon_4}C'''_{31}
%\end{align*}
%for some $C''_{31}$ and $C'''_{31}$. Since we have
%\begin{align*}
%& \epsilon+2\epsilon_2-s-t_1+t_2+\epsilon_4 \\
%&\leq (s-t_2+t_1)/4+(s-t_2+t_1)/4-s-t_1+t_2+(s-t_2+t_1)/4\\
%& \leq -(s-t_2+t_1)/4,
%\end{align*}
%$ C_{31}(i) \leq C''_{31}+C'''_{31}$ for $i\geq 1$.
%Therefore we confirmed that $ C_{31}(i)<\max(C'_{31},
% C''_{31}+C'''_{31})$ which does not depend on $i$.
\end{proof}


\section{Admissibility}
\label{sec:admissibility}
In this section, we give a sufficient condition for admissibility of the
generalized Bayes estimator with respect to a elliptically symmetric
prior density 
$g(\theta)=G(\| \theta \|_d)$.
The assumptions on $G$ are the following.

\begin{description}
\item[G1]  
$G(\eta)$ is continuously differentiable in $\eta>0$.
There exist $r_1 > 0$, $ t_1 \le t_2$ such
  that $ t_1\leq \eta G'(\eta)/G(\eta) \leq t_2$
 for all  $\eta \ge r_1$. 
\item[G1']  
By redefining $r_1$ if necessary in {\bfseries G1}, 
$t_1$ and $t_2$ can be taken as same sign.
%(This implies that $G(\eta)$ is positive or negative for all
% $\eta \ge r_1$.)
\item[G1'']
$G'(\eta)$ is continuously differentiable in $\eta>0$.
there exist 
$ t_3 \le t_4$ such  that
$ t_3 \leq \eta G''(\eta)/G'(\eta) \leq t_4$
 for all  $\eta \ge r_1$. 
\item[G2] 
$\int_1^\infty \eta^{p-1}G(\eta)d\eta=\infty$ and there exists
$ 0<\gamma \leq 2$ such that 
$\int_1^\infty \eta^{p-1}G(\eta)H_1^\gamma(\eta)d\eta<\infty$.
%$ g(\theta)$ is elliptically symmetric with respect to the
%	   origin, that is,
% where $ \| \theta \|_d^2= \sum d_i \theta_i^2  $ and 
% $ d_1 \geq \dots \geq d_p=1$.
\item[G3] $\lim_{\eta \to 0}\eta G'(\eta)/G(\eta)=t_0(>1-p). $
% \item[G2]  $ \int_0^1 \eta^{p-1}G(\eta)d\eta < \infty$ and %
%$\int_1^\infty \eta^{p-1}G(\eta)d\eta=\infty$, that is, improperness of
%	   $g$ occurs only at infinity.
%\item[G3] $ G(\eta) \leq  \eta^{2-p} (\int_\eta^\infty
%\beta(r)dr)^2/\{\eta\beta(\eta)\} $ for
%$ \eta \geq 1 $, where $ \beta(r) $ is given in Section \ref{sec:rv}.
%%\item[G4] $G$ is twice differentiable,
%\item[G4] $ \int_0^1 \eta^{p-1}|G'(\eta)|d\eta<\infty$ .
%\item[G5] There exist $r_1 > 0$, $ t_1 \le t_2$, $ t_3 \le t_4$, such
%  that 
%$ t_1\leq \eta G'(\eta)/G(\eta) \leq t_2$ and
%$ t_3\leq \eta G''(\eta)/G'(\eta) \leq t_4$ for all
%	    $\eta \ge r_1$. 
\item[FG1] $ \int_0^\infty r^{p-1}f(r)G(r)dr < \infty$ and $ \int_0^\infty
	   r^{p-2}F(r)G(r)dr < \infty$.
\end{description}

We discuss some implications of these assumptions.
% {\bfseries G3} becomes $ G(\eta) \le \eta^{2-p}\prod_{i=1}^n
%\mbox{Log}_i(\eta+c)$ for $ \beta $ given by \eqref{beta1}.
%The very mild assumption {\bfseries FG1} is just for finiteness of our 
%generalized Bayes estimators given by \eqref{bayes}.
By the assumption {\bfseries G3}, 
\begin{align*}
 \int_0^1 \eta^{p-1}G(\eta)d\eta<\infty \mbox{ and }
 \int_0^1 \eta^{p-1}|G'(\eta)| d\eta<\infty.
\end{align*}
 From the former
integrability and
{\bfseries G2}, the improperness of
   $g$ occurs only at infinity.
By 2 of Theorem \ref{thm-beta-H} and the assumption {\bf G2}, 
$ g(\theta)H_i^\gamma(\|\theta\|_d)$ for any fixed $i$ is integrable and hence
becomes a proper probability density by standardization.
Since $ H_i(\cdot)$ approaches $1$ as $i \to \infty$,
$ g(\theta)H_i^\gamma(\|\theta\|_d)$ 
is a sequence of proper densities approaching $g(\theta)$,
which is essential for using Blyth' method.

By {\bfseries G1} and 
Lemma \ref{lem:b3}, $G(\eta) = O(\eta^{t_2})$. Therefore if $t_2 < -p$,
then $g(\theta)=G(\|\theta\|_d)$ is a proper prior.  Since we are
considering an improper $g(\theta)$, we assume $t_2 \ge -p$ from now on.
Moreover $\eta^{t_1}/G(\eta)=O(1)$ by the assumption {\bfseries G1}.
Since $H_1 \in \mbox{RV}_{-1}$, $ \eta^{p-1}\eta^{t_1}H_1^2(\eta) $ for
$ t_1>2-p$ is not integrable
at infinity
and $ \eta^{p-1}G(\eta)H_1^2(\eta)$ is  not so either.
Hence we also assume $t_1 \leq 2-p$.
We now discuss when 
the integrability of $\int_1^\infty
\eta^{p-1}G(\eta)H_1^\gamma(\eta)d\eta $ in {\bfseries G2} holds
and the relationship with {\bfseries G1}.
When we take $ \beta(\eta)$ as in \eqref{beta1}, 
we easily see that there exists $L_1$ such that
$ H_1(\eta)\leq L_1 (\eta+c)^{-1} \{\log(\eta+c)\}^{-1}$.
If $t_2 \leq 2-p$, there clearly exists $L_2$ such that
$ G(\eta) \leq L_2\eta^{2-p}$ for $ \eta>1$.
Hence
\begin{align*}
\int_1^\infty \eta^{p-1}G(\eta)H^2_1(\eta)
d\eta & \leq 
 L_1^2 L_2 
 \int_1^\infty (\eta+c)^{-1}  \{\log(\eta+c)\}^{-2} d\eta \\
& = L_1^2 L_2  \{\log(1+c)\}^{-1}<
\infty ,
%.\label{A}
\end{align*}
which shows that there exists $\gamma(=2)$ for the integrability in 
{\bfseries G2}.
On the other hand, if $ t_2>2-p$ then the integrability in 
{\bfseries G2} may not be apparent. But the condition 
$t_2 \le 2-p$ is not a necessary
condition for the integrability.
%
%First we notice $ $
%If $ t_2 \le 2-p$, 
%\begin{align*}
% \int_1^\infty
%\eta^{p-1}G(\eta)H_1^\gamma(\eta)d\eta 
%\end{align*}
%That is, if $t_2 \le 2-p$ in {\bfseries G1}
%then the integrability in {\bfseries G2} is guaranteed.
%If $ t_2<2-p$, we can take $ \epsilon>0$ and $ \gamma=t_2+p+\epsilon$ 
%so that  $ 0< \gamma<2$ and 
%$\int_1^\infty \eta^{p-1}G(\eta)H_1^\gamma(\eta)d\eta<\infty$.
%If $ t_2=2-p$, the integrability is guaranteed for $\gamma=2$.
%For $t_2> 2-p$ the integrability may not apparent 
%An  understandable sufficient condition for
%$ \int_1^\infty \eta^{p-1}G(\eta)H_1^2(\eta)d\eta<\infty $
%is 
%%\begin{align}
%%G(\eta) \leq  \eta^{2-p} \left\{\int_\eta^\infty
%%\beta(r)dr\right\}^2/\{\eta\beta(\eta)\} \ \mbox{ for } \eta \geq 1,
%%\end{align}
%where $ \beta(r) $ is given in Section \ref{sec:rv} because


%Then in Theorem \ref{thm-zenkin}, $-t_1-a-p \le -t_1 - p \le t_2 - t_1$
%and 
%\[
%\max( -t_1-a-p,t_2-t_1)=t_2 - t_1.
%\]
%%In {\bfseries G3}, $\eta^{1-p} (\int_\eta^\infty
%%\beta(r)dr)^2/\beta(\eta) 
%%\in {\rm   RV}_{2-p}$.  
%% This implies that By {\bfseries G2} and {\bfseries G3},
%% $t_2$ should be restricted as $ p-2-\epsilon \leq t_2 \leq p+\epsilon $
%% for any $\epsilon>0$.  
%
If $G(\eta) $ is regularly varying, then for any $\epsilon>0$, we can
choose $r_1$, $t_1, t_2$, $t_3, t_4$,  such that $t_2-t_1<\epsilon $ and $
t_4-t_3<\epsilon$ for all $\eta \geq r_1$. 
However in ${\bf G1}$ we are allowing the case that
$\liminf_{\eta\rightarrow\infty} \eta G'(\eta)/G(\eta)$ is strictly less
than $\limsup_{\eta\rightarrow\infty} \eta G'(\eta)/G(\eta)$.  Hence we
are dealing with a broader class of $G(\eta)$ than the class of regularly
varying functions. It should also be noted that $ t_4 $ and $t_3$ are not
always smaller than $ t_2$ and $t_1$, respectively. See
\cite{geluk-dehaan1987} for the detail.

The generalized Bayes estimator $ \delta_{g}$ 
with respect to the improper density 
$g(\theta)$ is written as
\begin{align}
 \delta_g (x)&=\frac{\int_{R^p} \theta f(\|x - \theta\|)g(\theta)d\theta}
{\int_{R^p}  f(\|x - \theta\|)g(\theta)d\theta} 
\nonumber \\
&=x+ \frac{\int_{R^p} (\theta-x) f(\|x - \theta\|)g(\theta)d\theta}
{\int_{R^p}  f(\|x - \theta\|)g(\theta)d\theta} 
\nonumber \\
&=x+ \frac{\int_{R^p}  F(\|x - \theta\|) \nabla
 g(\theta)d\theta}
{\int_{R^p}  f(\|x - \theta\|)g(\theta)d\theta} \label{bayes}, 
\end{align}
which is well-defined if both $ \int_{R^p} F(\|x - \theta\|) \nabla
g(\theta)d\theta $ and $ \int_{R^p} f(\|x - \theta\|)g(\theta)d\theta $ are
integrable for all $x$.  These are guaranteed by the assumption {\bfseries
FG1} and Lemma \ref{thm-zenkin-1} in the below.  Write
\begin{align*}
 m(\psi | x) &= \int_{R^p} \psi(\theta) f(\|\theta - x\|)d\theta \\
M(\psi | x) &=  \frac{1}{C_f}\int_{R^p} \psi(\theta) F(\| \theta - x \|) 
 d\theta
\end{align*}
where $ C_f=\{\pi^{p/2}/\Gamma(p/2+1)\} \int_0^\infty z^{p+1}f(z)dz $.
%In the next section, we also consider the expectation 
%
%$E[\rho(Y)]$ where $Y \sim F(\|y-\theta\|)/C_f $ and 
%
Notice that $F( \cdot )/C_f$ is 
 a probability density function because
\begin{align*}
\int_{R^p} \|y - \theta \|^\alpha F(\|y - \theta \|)dy
& = \int_{R^p} \|y\|^\alpha \{ \int_{\| y  \|}s f(s)ds \} dy \\
&= c_p \int_0^{\infty} r^{p-1+\alpha} \int_{r}^\infty s f(s)ds dr \\
&= c_p \int_0^\infty r^{p+1+\alpha} \int_1^\infty t f(r t)dt dr \\
& = c_p \int_1^\infty t \{ \int_0^{\infty } r^{p+1+\alpha} f(r t ) dr \} dt \\
& = c_p \int_1^\infty t^{-p-1-\alpha}dt \cdot \int_0^\infty z^{p+1+\alpha}f(z)dz \\
& =\frac{c_p}{p+\alpha} \int_0^\infty z^{p+1+\alpha}f(z)dz.
\end{align*}
%where $c_p=2\pi^{p/2}\Gamma(p/2)$.  
%Therefore we require more
%restricted $ s>  2+\max(1, -t_1-a-p,t_2-t_1,t_2+a)$ to obtain  corresponding
%results of Theorem \ref{thm-zenkin} in the case where $Y \sim
%F(\|y-\theta\|)/C_f$.
Then $\delta_g$ is written as
\[
\delta_g (x)= x + C_f \frac{M(\nabla g|x)}{m(g|x)}.
\]
Note that by {\bf G1} the $j$-th element of $\nabla g$ is given by
\[
\nabla_j g(\theta)= d_j^2 \theta_j \frac{G'(\|\theta\|_d)}{\|\theta\|_d}.
\]
We also write
\[
h_i(x)=H_i(\|x \|_d).
\]

Now we state the following lemma in preparation of our main theorem.

\begin{lemma} 
\label{thm-zenkin-1}
\begin{enumerate}
\item 
Assume {\bf G1}, {\bfseries G3} and {\bf F1}.
Then 
\begin{align}
 m( g|x) & \approx g(x) & \mbox{ if }s &> \max(1, -t_1-p, t_2) \label{y1}\\
 m( gh_i^\gamma|x) & \approx g(x)h_i^\gamma(x) & \mbox{ if }s&> \max(1,
 \gamma-t_1-p, t_2) \label{y2}\\ 
% M( g|x) & \approx g(x) & \mbox{ if }s&> 2+\max(1, -t_1-p, t_2) \label{y3}\\
 M( gh_i^\gamma|x) & \approx g(x)h_i^\gamma(x) & \mbox{ if }s &> 2+\max(1,
 \gamma-t_1-p, t_2) \label{y3}\\ 
 M( g\|\theta\|^{-1}_d|x) & \approx g(x)\|x\|^{-1}_d & \mbox{ if }s &> 2+\max(1,
 1-t_1-p, t_2-1) \label{y4} \\
 M( gh_i^\gamma \|\theta\|^{-1}_d|x) & \approx g(x)h_i^\gamma(x)
\|x\|^{-1}_d & \mbox{ if }s &> 2+\max(1,
 \gamma+1-t_1-p, t_2-1) \label{y5}. 
\end{align}
\item Assume {\bf G1'}, {\bfseries G1''}, {\bfseries G3} and {\bf F1}.
Then
\begin{align}
M(\nabla_j g |x) & \approx
\nabla_j g (x)  & \mbox{ if }s &> 2+\max(1, -t_3-p, t_4) \label{y6}\\
M(\nabla_j gh_i^\gamma |x) & \approx
\nabla_j g (x)h_i^\gamma(x)  & \mbox{ if }s &> 2+\max(1, \gamma-t_3-p,
 t_4).  \label{y7}
\end{align}
\item $ M(\nabla g|x)/m(g|x)$ is bounded in $x$ if
$s>2+\max(1,1-t_1-p,t_2-1)$.
%\begin{align}
%\|x\|_d^{\epsilon-1}\left|   \right|& < C_b \times  G'(\|x\|_d) /\|x\|_d
%\label{z1}\\
%\|x\|_d^{\epsilon-1}\left|  M(\nabla_j g h_i^\gamma|x) - 
%\nabla_j g (x) h_i^\gamma (x) \right|& < C_5 \times  G'(\|x\|_d) h_i^\gamma(x)/\|x\|_d
%\label{z3}
%\end{align}
%are simultaneously satisfied 
% \begin{align}
% \|x\|_d^{\epsilon-1}\left|  \frac{M(\theta_j G'(\| \theta \|_d)
% H_i(\|\theta\|_d)/
% \| \theta \|_d)}{ G'(\|x\|_d)H_i^\gamma(\|x\|_d)/\|x\|_d}-  x_j 
% \right|& < C_3 \label{z3}\\
% \|x\|_d^{\epsilon-1} \left|  \frac{M(\theta_j G'(\| \theta \|_d)/\|
%  \theta \|_d)} 
% { G'(\|x\|_d)/\|x\|_d}-  x_j
% \right|& < C_1 \label{z1}\\
% \|x\|_d^{\epsilon}\left|  \frac{M( G(\| \theta \|_d)
% H_i(\|\theta\|_d))}{ G(\|x\|_d)H_i^\gamma(\|x\|_d)}-  1 
% \right|& < C_5
%  \label{z5}\\
% \|x\|_d^{\epsilon}\left|  \frac{m( G(\| \theta \|_d)
% H_i(\|\theta\|_d))}{ G(\|x\|_d)H_i^\gamma(\|x\|_d)}-  1 
% \right|& < C_4  \label{z4}\\
% \| x\|_d^\epsilon \left|  \frac{m( G(\|\theta\|_d))}{G(\|x\|_d)}-  1
% \right|& < C_2 \label{z2}
% \end{align}
%are satisfied simultaneously 
%for all sufficiently large $\|x\|_d$. 
%Furthermore $C_2, C_3, C_5$ do not depend on $i$.
\end{enumerate}
\end{lemma}

\begin{proof}
When we consider the asymptotic behavior of $ M(\cdot|x)$, that is,
the expectation under the probability density $F(\|\theta - x\|)/C_f$,
we have only to substitute $s$ for $s-2$ in order to have corresponding 
results for Theorem \ref{thm-zenkin} and Corollary \ref{th2.2coro}
because under the assumption {\bfseries F1} there exists $L_1$ such
that $r^{p+s-2}F(r)<L_1$ for all $r\ge r_0$.
We easily see that
\eqref{y1}, \eqref{y4} and \eqref{y6} follow from 
Theorem \ref{thm-zenkin} and that \eqref{y2}, \eqref{y3}, \eqref{y5} 
and \eqref{y7} follow from
 Corollary \ref{th2.2coro}.

Note that by the assumptions {\bfseries G1} and {\bfseries G3}
there exists $L_2$ such that $ |\eta G'(\eta)/G(\eta)|<L_2$
 for all $ \eta>0$.
Then $ M(\nabla g|x)/m(g|x) \leq d_1 M(|G'| \, |x)/m(g|x)
\leq L_2d_1 M(G/ \| \theta\| \, |x)/m(g|x) $.
The value at $x=0$ is clearly bounded under the assumption {\bfseries
 FG1} and the value at $\|x\|_d \to
\infty$  is bounded by \eqref{y1} and
\eqref{y4} if $s> 2+\max(1, 1-t_1-p, t_2-1)$.
\end{proof}

By part 3 of Lemma \ref{thm-zenkin-1}, 
$ M(\nabla g|x)/m(g|x)$ is bounded in $x$ and hence
the risk function of $\delta_g$
is finite  because 
\begin{align*}
 R(\theta,\delta_g)&=E[\|X-\theta+C_f M(\nabla g|X)/m(g|X)\|_Q^2]\\
&\leq Q_{\mathrm{max}} E[\|X-\theta+C_f M(\nabla g|X)/m(g|X)\|^2]\\
&\leq 2 Q_{\mathrm{max}} \, \{ E[\|X-\theta\|^2]+C_f^2 E[\|M(\nabla
 g|X)/m(g|X)\|^2]\},
\end{align*}
where $Q_{\mathrm{max}}$ is the largest eigenvalue of $ Q$.

Now we state the main theorem of this paper.
\begin{thm}
\label{thm:main}
\begin{enumerate}
 \item 
Assume {\bf G1}, {\bfseries G2}, {\bfseries G3}, {\bf F1}, {\bf FG1}.
Then 
the generalized Bayes estimator with respect to $g$ is admissible
 if 
$ s> 2+\max(1, 1+\gamma-t_1-p)$ and $t_2<2-p$.
\item
We also assume {\bfseries G1'} and {\bfseries G1''}. 
Then 
the generalized Bayes estimator with respect to $g$ is admissible
 if 
$ s> 2+\max(1, 3-t_1-p, t_2, 2-t_3-p, t_4)$ and 
$t_2 \ge 2-p$.
\end{enumerate}
%$ s> \max(3, 2+t_2 -t_1, 2+t_2) $ and $t_2< 2-p $ or if
%$ s> \max(3, 2+t_2-t_1, 2+t_4-t_3, 2+t_2, 4-p-t_3, t_4+1)$ 
%and $t_2 \geq 2-p$. 
\end{thm}
%A harmonic prior $ \|\theta\|^{2-p} $ and a slightly thicker 
%prior $ \| \theta \|^{2-p}\log(\|\theta\|+2) $ typically lead admissible
%estimators.
Although the moment conditions for $s$ in the theorem above 
looks complicated, 
it is just from the assumptions {\bfseries G1}, {\bfseries G1'} and
{\bfseries G1''} which make our class of
$G$ broader than the class of regularly varying functions.
We see that
the condition reduces to $s>3$ for regularly varying functions $G$.
Before giving a proof of the main theorem we present it as a 
corollary.
\begin{corollary} \label{main-coro-2}
Suppose that $G(\eta)$ is regularly varying with index $k$ for
$-p \le k \le 2-p$.
Assume {\bfseries G3}, {\bfseries F1} with $ s> 3$ and {\bfseries FG1}.
\begin{enumerate}
 \item Assume  $-p \le k < 2-p$ and 
$ G(\eta)$ is continuously differentiable.
  Then the generalized Bayes estimator with respect to
$g$ is admissible.
\item Assume $k=2-p$ and $ G(\eta)$ satisfies 
\begin{align}
\textstyle G(\eta) \leq  \eta^{2-p} \{\int_\eta^\infty
\beta(r)dr\}^2\{\eta\beta(\eta)\}^{-1} \ \mbox{ for } \eta \geq 1, \label{girigiri}
\end{align}
 and 
 is twice continuously differentiable. 
  Then the generalized Bayes estimator with respect to
$g$ is admissible.
\end{enumerate}
\end{corollary}
\begin{proof}
When $ -p \le k < 2-p $, 
we can take $t_1=k-\epsilon/3$ and $ t_2=k+\epsilon/3$ for 
any $ 0<\epsilon<\{3/2\}(2-p-k)$ in {\bfseries G1}. 
Let $\gamma=k+p+2\epsilon/3$. Clearly 
$ 0<\gamma<2$ and it satisfies the integrability in {\bfseries G2}.
Since $ t_2<2-p$, we have only to apply part 1 of Theorem \ref{thm:main}.
$  1+\gamma-t_1-p = 1+\epsilon$ and hence
the moment condition for $ k<2-p$ is $s>3$.

Next we consider the case $k=2-p$. 
Note that there exists $ L_1$ such that $ H_1(\eta)\leq L_1 \beta(\eta)/
{\int_{\eta}^\infty \beta(r)dr}$ by part 2 of Lemma
2.1. So if $G(\eta)$ satisfies
\eqref{girigiri}, we have
\begin{align*}
%\int_{\|\theta\|_d \ge 1} g(\theta)h^2_1(\theta)
\int_1^\infty \eta^{p-1}G(\eta)H_1^2(\eta)d\eta
 & \leq 
L_1 \int_1^\infty \beta(\eta)d \eta<\infty,
%
%\int_{\|\theta\|_d \ge 1} g(\theta)
%\left\{\frac{\beta(\|\theta\|_d)}
%{\int_{\|\theta\|_d}^\infty \beta(r)dr}\right\}^2
%d\theta \nonumber \\
%& \le L_1
%\int_{\|\theta\|_d \ge 1} \| \theta\|_d^{1-p}
%\beta(\|\theta\|_d)d\theta< c_{p,d}L_1 \int_1^\infty \beta(r)dr<
%\infty .
%.\label{A}
\end{align*}
which shows that the integrability in the assumption {\bfseries G2}
is guaranteed.
We can take $t_1=2-p-\epsilon$, $ t_2=2-p+\epsilon$,
$t_3=1-p-\epsilon$ and $t_4=1-p+\epsilon$ for any $ \epsilon>0$
in the assumptions
 {\bfseries G1}, {\bfseries G1''}.
Since $ t_2 \geq 2-p$, we have only to apply part 2 
of Theorem \ref{thm:main}.
Since
\begin{align*}
 3-t_1-p=1+\epsilon, \quad t_2 \leq 1+\epsilon, \quad 2-t_3-p=1+\epsilon,
\quad t_4 \leq \epsilon
\end{align*}
the moment condition for $ k=2-p$ is $s>3$.
\end{proof}
In particular the boundary case in \eqref{girigiri} 
by taking $\beta(\eta)$ in
(\ref{beta1}) was the motivating one for this paper.
\begin{corollary} \label{main-coro}
Assume {\bfseries F1} with $ s> 3$ and {\bfseries FG1}.
  Then the generalized Bayes estimator with respect to
%$ \|\theta\|_d^{2-p}$ and with respect to 
$ \|\theta\|_d^{2-p} \prod_{i=0}^n{\rm Log}_i(\|\theta\|_d+c)$,
%$g(\theta)=\|\theta\|_d^{2-p} \prod_{i=1}^n{\rm Log}_i(\|\theta\|_d+c)$,
where $n$ is a nonnegative integer and ${\rm Log}_n(c)>0$, 
is admissible.
\end{corollary}
In the normal case, \cite{Brown1971}'s sufficient conditions for
admissibility and inadmissibility are known.  He showed that the
generalized Bayes estimator with respect to $g$ is admissible if
\begin{align}
\label{brown-condition}
 \int_{\|x\|_d>1} \|x\|_d^{2-2p}\{m(g|x)\}^{-1} dx
\end{align}
diverges and inadmissible if \eqref{brown-condition} converges.
By Lemma \ref{thm-zenkin-1}, we see that 
\begin{align*}
 (1/2)G(\|x\|_d)<m(g|x)<2G(\|x\|_d)
\end{align*}
for sufficiently large $\|x\|_d$
and hence that $G(\eta) \leq \eta^{2-p}\prod_{i=0}^n{\rm Log}_i(\eta+c)
$ leads to admissibility and 
$G(\eta) \geq \eta^{2-p}\prod_{i=0}^{n-1}{\rm Log}_i(\eta+c)
 {\rm Log}_n^2(\eta+c)$ leads to inadmissibility.
%In the normal case of  generalized Bayes estimators with respect to 
%$g(\theta)=\|\theta\|^b$, $b>2-p$, is known to be
%inadmissible. 
Therefore our sufficient condition in Theorem
\ref{thm:main} is very close to being necessary.

We also notice that the prior density suggested in Corollary 
\ref{main-coro} becomes $ |\theta| \prod_{i=0}^n{\rm Log}_i(|\theta|+c)$
and $ \prod_{i=0}^n{\rm Log}_i(\|\theta\|_d+c)$ for $p=1, \ 2$
respectively, which are thicker than the Lebesgue measure.
In the normal case, \cite{Brown1971} has already pointed it out.

Furthermore we indicate that our moment condition $s>3$ is very tight
because, as pointed out in \cite{Perng1970}, admissibility requires the
existence of moment one degree higher than what is needed for finite risk in 
various estimation problems.

Now we give a proof of Theorem \ref{thm:main}.

\begin{proof}[Proof of Theorem \ref{thm:main}]
Let $\delta_{gi}$ denote the Bayes estimator with respect to the proper
prior density $g(\theta)h_i^\gamma(\theta)$. Then the Bayes risk difference
of $ \delta_g$ and $ \delta_{gi}$ with respect to the density 
$ g(\theta)h_i^\gamma(\theta)$ is written as
\begin{align*}
 \Delta &= \int_{R^{p}} \left[
R(\theta,\delta_{g})-R(\theta,\delta_{gi}) \right]
 g( \theta)h_i^\gamma(\theta )d \theta \\
&= \int_{R^p} \int_{R^p} [ \| \delta_{g}- \theta \|_Q^{2}- \| \delta_{gi}- \theta
 \|_Q^{2}]f(\|x-\theta\|)g(\theta)h_i^\gamma(\theta)
 d\theta dx\\
&= \int_{R^p} \bigg\{ [ \| \delta_{g} \|_Q^{2}-\| \delta_{gi}\|_Q^{2}]
  \int_{R^p} f(\| x- \theta\|) g(\theta)h_i^\gamma(\theta)d\theta
\\ & \quad 
-2(\delta_{g}-\delta_{gi})Q'\int_{R^p} \theta f(\|x-\theta\|)
g(\theta)h_i^\gamma(\theta)
 d\theta \bigg\}dx \\
&= \int_{R^{p}} \| \delta_{g} - \delta_{gi} \|_Q^{2} \left\{\int_{R^p} 
 f(\|x-\theta\|)g(\theta)h_i^\gamma(\theta)
 d\theta\right\} dx \\
&= C_f^2\int_{R^{p}} \left\| \frac{M(\nabla g|x)}{m(g|x)} - 
\frac{M(\nabla \{gh_i^\gamma\}|x)}{m(gh_i^\gamma|x)}
\right\|_Q^2 m(gh_i^\gamma|x)dx \\
&= C_f^2\int_{R^{p}} \left\| \frac{M(\nabla g|x)}{m(g|x)} - 
\frac{M(\nabla g h_i^\gamma|x)}{m(gh_i^\gamma|x)}-\frac{M(g \nabla h_i^\gamma|x)}{m(gh_i^\gamma|x)}
\right\|_Q^2 m(gh_i^\gamma|x)dx .
\end{align*}

In the same way as in \cite{Brown-Hwang1983},
we have
\begin{align*}
 \Delta & \leq 2C_f^2 Q_{\mathrm{max}}
 \int_{R^{p}} \left\| \frac{M(\nabla g|x)}{m(g|x)} - 
\frac{M(\nabla g h_i^\gamma|x)}{m(gh_i^\gamma|x)}
\right\|^2 m(gh_i^\gamma|x)dx \\
& \qquad + 2 C_f^2Q_{\mathrm{max}}
\int_{R^{p}} \left\| \frac{M(g \nabla h_i^\gamma|x)}{m(gh_i^\gamma|x)}
\right\|^2 m(gh_i^\gamma|x)dx  \\
&=2C_f^2Q_{\mathrm{max}}(B_i+A_i). \qquad  \mathrm{(say)}
\end{align*}
%where $Q_{\mathrm{max}}$ is the largest eigenvalue of $ Q$.

Using the Cauchy-Schwartz inequality for $A_i$, we have
\begin{align*}
 A_i &=\gamma^2 \int_{R^{p}} \left\| M(g h_i^{\gamma-1} \nabla h_i|x)
\right\|^2 \{ m(gh_i^\gamma|x) \}^{-1} dx \\
& \leq \gamma^2 \int_{R^{p}} \frac{M(gh_i^\gamma|x)}{m(gh_i^\gamma|x)}
 M(g h_i^{\gamma-2} \| \nabla h_i \|^2|x)
 dx \\
& \leq \gamma^2 \int_{R^{p}} \frac{M(gh_i^\gamma|x)}{m(gh_i^\gamma|x)}
 M(g h_1^{\gamma-2} \| \nabla h_i \|^2|x)
 dx
\end{align*}
for $ 0<\gamma\leq 2$.
The ratio $M(gh_i^\gamma|x)/m(gh_i^\gamma|x)$ is bounded from above
by $M(g|x)/m(gh_1^\gamma|x)$ and hence the value at $x=0$ is clearly
bounded under the assumption {\bfseries FG1}.
By \eqref{y2} and \eqref{y3}, we have
\begin{align*}
\lim_{\|x\|_d \to \infty} \frac{M(gh_i^\gamma|x)}{m(gh_i^\gamma|x)}=1
\end{align*}
uniformly in $i$.  This implies that there exists $c_1$ such that
$M(gh_i^\gamma|x)/m(gh_i^\gamma|x)<c_1$ for all $x$ and for all $i$. Then
\begin{align*}
A_i & \leq \gamma^2c_1\int_{R^{p}}  M(g h_1^{\gamma-2} 
\| \nabla h_i  \|^2 |x) dx \\
& = \gamma^2 c_1 \int_{R^{p}}\{1/C_f\} F(\| x - \theta \|)dx 
\int_{R^{p}} g(\theta) 
h_1^{\gamma-2}(\theta) \| \nabla h_i (\theta)\|^2
 d\theta \\
&= \gamma^2 c_1 \int_{R^{p}} g(\theta) h_1^{\gamma-2}(\theta) 
\| \nabla h_i(\theta) \|^2 d\theta.
\end{align*}
By 4 of Theorem \ref{thm-beta-H} we have $  \| \nabla h_i(\theta) \|
< 2 d_1 \beta(\|\theta\|_d)/\int_{\|\theta\|_d}^\infty \beta(r)dr $
and together with 2 of Theorem \ref{thm-beta-H}
$ \| \nabla h_i(\theta) \|/h_1(\theta)$ for all $\theta$
is bounded from above by $L_3$ independent of $i$. Therefore
%\sup_i \int g(\theta) \| \nabla h_i(\theta) \|^2 d\theta &< 
\[
A_i \leq 
\gamma^2 c_1L_3^2 \int_{R^{p}} g(\theta) h_1^{\gamma}(\theta) 
d\theta,
\]
which is bounded by the assumption {\bfseries G2}.
Furthermore $ \| \nabla h_i(\theta)\| \to 0$ as $i \to \infty $ by 3 of
Theorem \ref{thm-beta-H}. Therefore by the dominated convergence theorem $
A_i $ converges to $0 $ as $i \to \infty$.


Next we consider $ B_i$.
$ M(\nabla g|x)$ and $M(\nabla g h_i^\gamma|x)$ at $x=0$ are zero vectors 
because $g$ and $h_i^\gamma$ are function of $ \| \theta \|_d$.
So the integrand of $B_i$ is bounded around $x=0$.
For the asymptotic property of the integrand of $B_i$,
we need to distinguish two cases: $t_2<2-p$ and $t_2\geq 2-p$.
When $t_2<2-p$, we can bound the norm in the integrand of $B_i$
from above somewhat roughly.
Using \eqref{y1}, \eqref{y2}, \eqref{y4} and \eqref{y5} in Lemma
 \ref{thm-zenkin-1} and noting
that $ |\eta G'(\eta)/G(\eta)|<L_2$ by the assumptions 
{\bfseries G1} and {\bfseries G3}
we have
\begin{align*}
 \frac{1}{d_j^2}\left| \frac{ M(\nabla_j g|x)}{m(g|x)}-\frac{ M(\nabla_j g
 h_i^\gamma|x)}{m(gh_i^\gamma|x)} \right| 
&  <  \frac{ M(|G'|\, |x)}{m(g|x)}
+\frac{ M(|G'|h_i^\gamma \, |x)}{m(gh_i^\gamma|x)}
\\
%-\frac{ M(\nabla_j g
% h_i^\gamma|x)}{M(gh_i^\gamma|x)}\frac{M(gh_i^\gamma|x)}{m(gh_i^\gamma|x)} \right| \\
&  <  L_2 \left(\frac{ M(G \|\theta\|^{-1}\, |x)}{m(g|x)}
+\frac{ M(G h_i^\gamma\|\theta\|^{-1} |x)}{m(gh_i^\gamma|x)} \right)\\
&  < c\|x\|_d^{-1}
\end{align*}
%\begin{align*}
%& \frac{1}{d_j^2}\left| \frac{ M(\nabla_j g|x)}{m(g|x)}-\frac{ M(\nabla_j g
% h_i^\gamma|x)}{m(gh_i^\gamma|x)} \right| \\
%&\qquad = \frac{1}{d_j^2}\left| \frac{ M(\nabla_j g|x)}{M(g|x)}
%\frac{M(g|x)}{m(g|x)}
%-\frac{ M(\nabla_j g
% h_i^\gamma|x)}{M(gh_i^\gamma|x)}\frac{M(gh_i^\gamma|x)}{m(gh_i^\gamma|x)} \right| \\
%& \qquad \leq 2\left| \frac{G'(\|x\|_d)}{G(\|x\|_d)} \right|
%+ 2\left| \frac{G'(\|x\|_d)}{G(\|x\|_d)} \right|\\
%&\qquad  < c\|x\|_d^{-1}
%\end{align*}
for all sufficiently large $\|x\|_d$ and for all $i$.
When $t_2 \geq 2-p$, we have to bound it from above more strictly.
By \eqref{y1}, \eqref{y2}, \eqref{y6} and \eqref{y7} in
Lemma \ref{thm-zenkin-1},  we have
\begin{align*}
& \frac{1}{d_j^2}\left| \frac{ M(\nabla_j g|x)}{m(g|x)}-\frac{ M(\nabla_j g
 h_i^\gamma|x)}{m(gh_i^\gamma|x)} \right| \\
&\qquad  = \left| \frac{G'(\|x\|_d)}{G(\|x\|_d)\|x\|_d}
\frac{x_j+O(\|x\|_d^{1-\epsilon_0}) }{1+O(\|x\|_d^{-\epsilon_0})} -
\frac{G'(\|x\|_d)}{G(\|x\|_d)\|x\|_d}
\frac{x_j+O(\|x\|_d^{1-\epsilon_0}) }{1+O(\|x\|_d^{-\epsilon_0})}
\right| \\
&\qquad  < c\|x\|_d^{-1-\epsilon_0} \\
&\qquad  < H_1(\|x\|_d) 
\end{align*}
for some $\epsilon_0$,
for all sufficiently large $\|x\|_d$ and for all $i$.
%where $c$ does not depend on $i$.
Moreover $m(gh_i^\gamma|x)\leq m(g|x)$ and $m(g|x) < 2G(\|x\|_d)$
for all sufficiently large $\|x\|_d$ by \eqref{y1}.  
Therefore there exist
$C_1$, $C_2$, $C_3$ and $C_4$ such that the integrand of $B_i$ is less
than
\begin{align*}
%\textstyle L(\|x\|_d)=
\begin{cases}
\min\{C_1, C_2 \|x\|_d^{-2+t_2}\} & \mbox{ if } t_2<2-p \\
\min\{C_3, C_4 G(\|x\|_d)H_1^2(\|x\|_d)\} & \mbox{ if } t_2 \ge 2-p.
\end{cases}
\end{align*}
%for all sufficiently large $\|x\|_d$. 
Therefore 
$ B_i $ converges to $0 $ as $i \to \infty$ by the dominated convergence
theorem.
%, in view of 
% \begin{align} \label{B}
%\[
% \int_0^\infty r^{p-1}
%%L(r)
%\min(C_1, C_2 r^{-2+t_2})dr<\infty, 
% \int_0^\infty r^{p-1}
%%L(r)
%\min(C_3, C_4 G(\|x\|_d)H_1^2(\|x\|_d) dr<\infty.
%\]
%\end{align}

Finally we confirm that we use \eqref{y1}--\eqref{y5} for $t_2<2-p$
and \eqref{y1}--\eqref{y3}, \eqref{y6} and \eqref{y7} for $ t_2 \geq
 2-p$. Note also $\max(1, t_2)=1$ for $t_2<2-p$ in the moment condition.
\end{proof}

\section{The generalized Bayes estimator with respect to
 the harmonic  prior and its minimaxity} 
In this section, we show that the generalized Bayes estimator with
respect to the harmonic prior has a form simple enough to check 
some sufficient conditions for minimaxity under the quadratic
loss function $L_I(\theta,\delta)=\| \delta- \theta \|^2$
given in early studies.
We demonstrate that it is minimax for some $f$.

In \eqref{bayes}, the generalized Bayes estimator can be also written as
\begin{align*}
 \delta_g(x)=x+C_f \frac{\nabla_x M(g|x)}{m(g|x)}.
\end{align*}
For $p \geq 3$ and $g(\theta)=\|\theta\|^{2-p}$, 
we have
%the marginal
% density function of $ X$ with respect to the harmonic prior
%is written as
\begin{align}
  m(g|x)&=\int_{R^{p}}f(\|x-\theta\|)\| \theta \|^{2-p}d\theta 
 =\int_{R^{p}}f(\|\eta\|)\| x-\eta \|^{2-p}d\eta \nonumber \\
 & \ \ = %\frac{2\pi^{p/2-1/2}}{\Gamma(p/2-1/2)}
c_{p-1}\int_{0}^{\infty}\int_{0}^{\pi}f(\lambda)(\lambda^2
+2\lambda r\cos \varphi +r^2)^{1-p/2}\lambda^{p-1}
\sin^{p-2}\varphi d\lambda d \varphi \nonumber \\
 & \ \ = %\frac{2 \pi^{p/2-1/2}}{\Gamma(p/2-1/2)}
c_{p-1} r^2 \int_{0}^{\infty}\int_{0}^{\pi}f(rt)
(1+2t\cos \varphi+t^2)^{1-p/2}t^{p-1}\sin^{p-2}\varphi 
dt d \varphi \nonumber \\
 & \ \ = %\frac{2\pi^{p/2}}{\Gamma(p/2)}
c_{p}\left(r^2\int_{0}^{1}t^{p-1}f(rt)dt+r^2\int_{1}^{\infty}
t f(rt)dt \right)
 \nonumber \\ 
& \ \ = %\frac{2\pi^{p/2}}{\Gamma(p/2)}
c_{p}\left( \left[ -t^{p-2} F(rt) \right]_0^1+
(p-2)\int_0^1 t^{p-3}F(rt)dt+ F(r)
 \right)
 \nonumber \\ 
 & \ \ = %\frac{4\pi^{p/2}}{\Gamma(p/2-1)}
c_p(p-2)\int_{0}^{1}t^{p-3}F(rt)dt,
\end{align}
where $ r=\|x\| $.
The fifth equality in the above equation follows from
the relation
% due to Erd\'{e}lyi {\it et al.}(1953),
\begin{equation*}
 \int_{0}^{\pi}(1+2t\cos \varphi +t^2)^{1-p/2}
\sin^{p-2}\varphi d\varphi=B(p/2-1/2,1/2)\min(t^{2-p},1),
\end{equation*}
which is proved in Lemma \ref{int} in the end of this section.
% The fourth equalty is from an integration by parts for the first term
% of the left hand side.
In the same way, we have
\begin{align*}
C_f \nabla_x M(g|x)=  -x c_p(p-2)\int_{0}^{1}t^{p-1}F(rt)dt.
\end{align*}
Hence 
%Moreover we have 
%\begin{align*}
%& \int_{R^{p}}\theta f(\|x-\theta\|)\| \theta \|^{2-p}d\theta \\
%& =\int_{R^{p}}(\theta -x) f(\|x-\theta\|)\| \theta \|^{2-p}d\theta 
%+x \int_{R^{p}} f(\|x-\theta\|)\| \theta \|^{2-p}d\theta \\
%&=
%\frac{\partial}{\partial x}\int F(\|x-\theta\|)
%\| \theta \|^{2-p}d\theta +x \int f(\|x-\theta\|)\| \theta
% \|^{2-p}d\theta \\ 
%&=  x c_p(p-2)\left( \int_{0}^{1}t^{p-3}F(rt)dt -
%\int_{0}^{1}t^{p-1}F(rt)dt\right),
%\end{align*}
%Therefore we have 
the generalized Bayes estimator is written as
$ \delta_{*}(X)=(1-\phi_{*}(\|X\|)/\|X\|^2)X $,  where 
\begin{align*}
 \phi_{*}(r)=r^2\frac{\int_{0}^{1}t^{p-1}F(rt)dt}
{\int_{0}^{1}t^{p-3}F(rt)dt}.
\end{align*}
Some properties of the behavior of $ \phi_{*}(r) $ are easily derived as
follows. 
\begin{thm} \label{lemma1}
 \begin{enumerate}
  \item $ \lim_{r \rightarrow \infty }\phi_{*}(r)
=(p-2)E_{0}(\|X\|^{2})/p $.
\item $ \phi_{*}(r) $ is  nondecreasing in $r $ for any $ f$. 
\item $ \phi_{*}(r)/r^2 $ is nonincreasing in $r $ 
if $ F(t)\{t^2f(t)\}^{-1} $ is nonincreasing.   
 \end{enumerate}
\end{thm}
\begin{proof}
%As the function 
$ \phi_{*}(r) $ can be written as
$  \int_{0}^{r}t^{p-1}F(t)dt/
\int_{0}^{r}t^{p-3}F(t)dt$
and we have
\begin{equation*}
 \lim_{r \rightarrow \infty }\phi_{*}(r)= 
\frac{\int_{0}^{\infty}t^{p-1}F(t)dt}
{\int_{0}^{\infty}t^{p-3}F(t)dt} 
=\frac{p-2}{p}\frac{\int_{0}^{\infty}t^{p+1}f(t)dt}
{\int_{0}^{\infty}t^{p-1}f(t)dt}=\frac{p-2}{p}E_{0}[\|X\|^{2}]. 
\end{equation*}
The derivative of $ \phi_{*}(r) $ is calculated as
\begin{align*}
 \phi'_{*}(r)=\frac{r^{p-3}F(r)}
{\left( \int_{0}^{r}t^{p-3}F(t)dt \right)^2}
\int_{0}^{r}(r^2-t^2)t^{p-3}F(t)dt,
\end{align*}
which is nonnegative for any $f$. The derivative of $ \phi_{*}(r)/r^2 $ is
 calculated as
\begin{align*}
 \frac{d}{dr}\left(\phi_{*}(r)/r^2 \right) & =
r \left( \int_{0}^{1}t^{p-3}F(rt)dt \right)^{-2} \left( 
 \int_{0}^{1}t^{p-1}F(rt)dt\int_{0}^{1}t^{p-1}f(rt)dt \right. \\
 & \qquad - \left.
\int_{0}^{1}t^{p+1}f(rt)dt\int_{0}^{1}t^{p-3}F(rt)dt \right).
\end{align*}
If $ F(t)\{t^2f(t)\}^{-1} $ is nonincreasing, 
the right-hand side of the equality above is nonpositive by the
 covariance  inequality.  
\end{proof}

Now we consider the minimaxity of $\delta_*$. 
%In our knowledge, 
We present a brief list of known sufficient conditions for minimaxity given
in previous papers, for the estimator of the form \eqref{orthogonal} with
nonnegative and nondecreasing $\phi(r)$.

\medskip
\footnotesize
\begin{center}
\begin{tabular}{|c|c|c|c|}\hline
Author & $ p $ & $ \phi(r)/r^2 $ & upper bound of $\phi$\\ \hline \hline
\multicolumn{4}{|c|}{general} \\ \hline
\cite{Berger1975} & $ p \geq 3 $ & &  
$ 2(p-2)\inf_{s \in U}F(s)/f(s)  $   \\ \hline
\cite{Brandwein1979} & $ p \geq 4 $ & $ \searrow $ & 
$ 2(p-2)(pE_{0}(\|X\|^{-2}))^{-1} $ \\ \hline \hline
 \multicolumn{4}{|c|}{unimodal or $ f $ is nonincreasing} \\ \hline
\cite{Brand-Straw1978} & $ p \geq 4 $ & $ \searrow $ &  
$ 2p((p+2)E_{0}(\|X\|^{-2}))^{-1} $ \\ \hline
\cite{Ralescu-etal1992} & $p=3 $ & $ \searrow $ &
$ 0.93(E_{0}(\|X\|^{-2}))^{-1} $ \\ \hline \hline
 \multicolumn{4}{|c|}{$ F(t)/f(t) $ is nondecreasing} \\ \hline
\cite{Bock1985}  & $ p \geq 4 $ & $ \searrow $ & 
$ 2(E_{0}(\|X\|^{-2}))^{-1}$ \\ \hline
\hline
 \multicolumn{4}{|c|}{scale mixtures of multivariate normal} \\ \hline
\cite{Strawderman74}   & $ p \geq 3 $ & $ \searrow $ & 
$ 2(E_{0}(\|X\|^{-2}))^{-1}$ \\ \hline
\end{tabular}
\end{center}
\normalsize
\medskip
In the table, $ U=\{t \geq 0| f(t)>0 \}$ and an arrow $ \searrow $ means {\it
nonincreasing}. 
It is noted that 
$ f(t) $ is nonincreasing in $t $ if $ F(t)/f(t) $ is nondecreasing in 
$t $ and that $ F(t)/f(t) $ is nondecreasing in $t $ if $f$ is a scale 
mixtures of multivariate normal.

Combining Theorem \ref{lemma1} and  the table above, 
we can derive a sufficient condition for minimaxity of $ \delta_{*}(X) $
and we state it in the following theorem for $p \geq 4$.

\begin{thm}
\label{thm:minimaxity}
\begin{enumerate}
\item Assume $ t^{-2}F(t)/f(t)$ is nonincreasing.
\begin{enumerate}
\item $\delta_*$ is minimax if $ E_0[\|X\|^2]E_0[\|X\|^{-2}]
\leq 2$.
\item Assume also $ f(t)$ is nonincreasing.
Then $\delta_*$ is minimax \\ if $ (p^2-4)E_0[\|X\|^2]E_0[\|X\|^{-2}]
\leq 2p^2$.
 \item Assume also $ F(t)/f(t)$ is nondecreasing.
Then $\delta_*$ is minimax \\ if $ (p-2)E_0[\|X\|^2]E_0[\|X\|^{-2}]
\leq 2p$.
\end{enumerate}
\item Assume $0<\inf_{s \in U}F(s)/f(s)<\infty$.
Then $\delta_*$ is minimax \\ if $ E_0[\|X\|^2]\leq 2p\inf_{s \in
 U}F(s)/f(s)$. 
\end{enumerate}
\end{thm}
\cite{Berger1975} and \cite{Bock1985} gave several examples
of $f$, checked the monotonicity of $f(t)$, $F(t)/f(t)$, and
$t^{-2}F(t)/f(t)$ and  calculated an upper bound of $\phi(r)$.
In this paper we give just two examples but we believe that
the estimator $ \delta_{*}(X) $ is  minimax for a broad class of 
spherically symmetric distributions.
\begin{example}
%{\bf Example 1.} \ 
We consider $  f(s)=s^{\alpha}\exp(-\beta s^2) $ for $ \alpha,\ \beta>0 $. 
We have
\begin{equation*}
 \frac{F(t)}{t^2f(t)}=\int_{1}^{\infty}u^{\alpha+1}\exp(\beta t^2 (1-u))du,
\end{equation*}
which is decreasing in $ t $. By an integration by parts, we have
\begin{align*}
 \frac{F(t)}{f(t)}&=\frac{1}{2\beta}+\frac{\alpha}{2\beta}
\frac{\int_t^\infty s^{\alpha-1}\exp(-\beta s^2)ds}{t^\alpha
\exp(-\beta t^2)}\\
&=\frac{1}{2\beta}+\frac{\alpha}{2\beta}\int_1^\infty
u^{\alpha-1}\exp(\beta t^2\{ 1-u^2\})du
\end{align*}
and hence 
%A simple calculation gives 
$ \inf_{t \geq 0}F(t)/f(t)=(2\beta)^{-1} $.
We also have $ E_{0}(\|X\|^{2})
=(p/2+\alpha/2)/\beta $ and $E_{0}(\|X\|^{-2})^{-1}=(p/2+\alpha/2-1)/\beta $.
Therefore the generalized Bayes estimator is minimax 
if $ \alpha \leq p $ for $ p \geq 3 $ by \cite{Berger1975}'s conditions 
and if $ \alpha \geq 4-p $ for $ p \geq 4 $ by \cite{Brandwein1979}'s
 conditions regardless of $ \beta $. 
Hence the estimator for $ p \geq 4 $ is minimax regardless of $ \alpha $
 and $ \beta$.
\end{example}

%Hence we have a following result.
%\begin{theorem}
%We assume that $  f(s)=s^{\alpha}\exp(-\beta s) $ for $ \alpha,\ \beta>0 $.
%The generalized Bayes estimator is minimax if $ p \geq 4 $ or if 
%$ \alpha \leq 3/2 $ for $ p=3 $.
%\end{theorem}
%the estimator for $ p \geq 4 $ is minimax.
%it is minimax if 
%$ p \geq 4 $ and if $ \alpha \leq 3/2 $ for $ p=3 $  
%%%%%\end{example}
%Next we consider 
%$  f(t)=\exp(-t/2)+a\exp(-t/(2b)) $ for $ a>0 $, $ b>1  $.  
%
%We have
%\begin{equation*}
% \frac{F(t)}{tf(t)}=t^{^1}\frac{1+ab\exp(t/2-t/(2b))}{1+a\exp(t/2-t/(2b))},
%\end{equation*}
%which is decreasing in $t$ if $ a \geq b^{-1}(b-1)\exp(-2b(b-1)^{-1})$.
%\begin{lemma}
%A function  $ s^{-1}(1+ab\exp(s))/(1+a\exp(s)) $ for $ a>0 $, $ b>1  $
% is decreasing in $s$ if $ b>1  $ and 
%$ a \geq b^{-1}(b-1)\exp(-2b(b-1)^{-1})$. 
%\end{lemma}
%\begin{proof}
%The numerator of the derivative of the function is calculated as
%\begin{eqnarray*}
% & &s(1+a\exp(s))ab\exp(s)-(1+ab\exp(s))(1+a\exp(s)+as\exp(s)) \\
%& & \quad =a(b-1)s\exp(s)-a(b+1)\exp(s)-a^{2}b\exp(2s)-1 \\
%& & \quad =\exp(2s)\left[  a(b-1)s\exp(-s)-a(b+1)\exp(-s)-a^{2}b \right] -1 \\
%& & \quad \leq \exp(2s)\left[ a(b-1)\exp(-2b(b-1)^{-1})-a^{2}b \right] -1 \\
%& & \quad \leq 0.
%\end{eqnarray*}
%\end{proof}

%{\bf Example 2.} \ 
\begin{example}
We consider 
$  f(t)=\exp(-t^2/2)-a\exp(-t^2/\{2b\}) $ for $ 0< a \leq 1 $, $ 0<b<1 $. 
Note that if $ a \leq b $ then $f $ is unimodal and if $ a > b $ then 
$ f $ is not. We  easily see that 
%$ F(t)/f(t) $ is monotone
% nonincreasing, that 
$ \inf_{t \geq 0} F(t)/f(t)=1 $ and that
$E_{0}[\|X\|^{2}]=p(1-ab^{p/2+1})/(1-ab^{p/2}) $.
%\begin{equation*}
% E_{0}[\|X\|^{2}]=p\frac{1-ab^{p/2+1}}{1-ab^{p/2}}.
%\end{equation*}   
Because $ (1-ab^{p/2+1})/(1-ab^{p/2}) \leq 2 $ for 
 $ 0< a \leq 1 $, $ 0<b<1 $, the generalized Bayes estimator is minimax
 by \cite{Berger1975}.
\end{example}

The following lemma is stated in a more general form in 3.036 of
\cite{Gradshteyn-Ryzhik}, but it is incorrectly stated with an errata
posted on the book's web page.  Maruyama pointed out this error and he is
acknowledged in the errata for 3.036.  Since a derivation of the formula
is not easily accessible, we provide our own proof.

%\begin{thm}
\begin{lemma}
 \label{int}
For $ \alpha > -1/2 $ and $|a|<1 $,
 \begin{equation*}
  \int_{0}^{\pi}
(1+2a \cos \varphi+a^{2})^{-\alpha}\sin^{2\alpha}\varphi d\varphi=
B(\alpha+1/2,1/2).
 \end{equation*}
%\end{thm}
\end{lemma}

\begin{proof}
 Let $ g(\varphi)=(1+2a \cos \varphi+a^{2})^{-1}\sin^{2}\varphi $.
Then we have the derivative 
\begin{equation*}
 g'(\varphi)=2 \sin \varphi \frac{(a\cos \varphi +1)(\cos \varphi +a)}{(1+2a \cos \varphi+a^{2})^{2}}.
\end{equation*} 
We see that $ g(\varphi) $ is monotone increasing 
from $ g(0)=0$ to $ g(\arccos (-a))=1 $ and decreasing from
 $g(\arccos (-a))=1 $ to $ g(\pi)=0 $.
Therefore we have
\begin{align}
& \int_{0}^{\pi} (1+2a \cos \varphi+a^{2})^{-\alpha}\sin^{2\alpha}\varphi d\varphi \nonumber \\
& =\left( \int_{0}^{\arccos(-a)}+\int_{\arccos(-a)}^{\pi}\right)
  (1+2a \cos \varphi+a^{2})^{-\alpha}\sin^{2\alpha}\varphi d\varphi \nonumber \\
&=\int_{0}^{\arccos(-a)} (1+2a \cos \varphi+a^{2})^{-\alpha}\sin^{2\alpha}\varphi d\varphi
\nonumber \\
& \quad +\int_{0}^{\arccos(a)} (1-2a \cos \rho+a^{2})^{-\alpha}\sin^{2\alpha}\rho d\rho \nonumber \\
&  = \int_{0}^{1} t^{\alpha}(d\varphi/dt)dt+\int_{0}^{1} s^{\alpha}(d\rho/ds)ds,
 \label{h} 
\end{align}
where $ t=(1+2a \cos \varphi+a^{2})^{-1}\sin^{2}\varphi $ and 
$ s=(1-2a \cos \rho+a^{2})^{-1} \sin^{2}\rho $.
Here $(d\varphi/dt) $ and $ (d\rho/ds) $ are calculated as
\begin{align*}
 d\varphi/dt &=\frac{1}{2tA(t)}\left( 1-\left\{at-A\left(t
 \right)\right\}^2 
\right)^{1/2}\\
 d\rho/ds &=\frac{1}{2sA(s)}\left( 1-\left\{as+A\left(s
 \right)\right\}^2 \right)^{1/2},
\end{align*}
where $ A(t)=(1-t)^{1/2}(1-a^{2}t)^{1/2} $. 
Let
\begin{equation*}
h(t)=\frac{1}{2tA(t)}
\left\{\left( 1-\left\{at-A\left(t \right)\right\}^{2}
\right)^{1/2}+\left( 1-\left\{at+A\left(t \right)\right\}^{2}
\right)^{1/2}\right\}.
\end{equation*}
Then we have
$
h^{2}(t)=\{2tA(t)\}^{-2}\{2-2a^{2}t^{2}-2A(t)^{2}+2B(t)\}
$, 
where
\begin{align*}
 B(t) & =\left( 1-\{at-A(t)\}^{2}-\{at+A(t)\}^{2}
+\{a^{2}t^{2}-A^{2}(t)\}^2\right)^{1/2}\\
& = t(1-a^{2}),
\end{align*}
which implies  $ h(t)=t^{-1/2}(1-t)^{-1/2} $. 
Therefore we get
\begin{equation*}
 \mathrm{the \ right \ hand \ side \ of \
  }\eqref{h}=\int_{0}^{1}t^{\alpha}h(t)dt 
=B(\alpha+1/2,1/2).
\end{equation*}
\end{proof}


\bibliographystyle{ims}
%\bibliographystyle{acmtrans-ims}
\bibliography{arxiv-bib}

\end{document}

% LocalWords:  minimaxity Yuzo Maruyama Akimichi Takemura Kashiwanoha Kashiwa
% LocalWords:  shi Hongo Bunkyo ku Strawderman Strawderman's Baranchik dt ds tx
% LocalWords:  eqn dr L'Hospital bibun HH Casella TPE sekibun hyouka dx zenkin
% LocalWords:  af nonincreasing Ld eq coro FG Blyth dy gh gi bayes nonpositive
% LocalWords:  pE du tA sA

arxiv-admissibility.bbl
\begin{thebibliography}{25}
\expandafter\ifx\csname natexlab\endcsname\relax\def\natexlab#1{#1}\fi
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi

\bibitem[{Baranchik(1970)}]{Baranchik1970}
\textsc{Baranchik, A.~J.} (1970).
\newblock A family of minimax estimators of the mean of a multivariate normal
  distribution.
\newblock \textit{Ann. Math. Statist.} \textbf{41} 642--645.

\bibitem[{Berger(1975)}]{Berger1975}
\textsc{Berger, J.} (1975).
\newblock Minimax estimation of location vectors for a wide class of densities.
\newblock \textit{Ann. Statist.} \textbf{3} 1318--1328.

\bibitem[{Berger(1976)}]{Berger1976b}
\textsc{Berger, J.~O.} (1976).
\newblock Admissible minimax estimation of a multivariate normal mean with
  arbitrary quadratic loss.
\newblock \textit{Ann. Statist.} \textbf{4} 223--226.

\bibitem[{Bingham et~al.(1987)Bingham, Goldie and Teugels}]{BGT1987}
\textsc{Bingham, N.~H.}, \textsc{Goldie, C.~M.} and \textsc{Teugels, J.~L.}
  (1987).
\newblock \textit{Regular variation}, vol.~27 of \textit{Encyclopedia of
  Mathematics and its Applications}.
\newblock Cambridge University Press, Cambridge.

\bibitem[{Blyth(1951)}]{Blyth1951}
\textsc{Blyth, C.~R.} (1951).
\newblock On minimax statistical decision procedures and their admissibility.
\newblock \textit{Ann. Math. Statist.} \textbf{22} 22--42.

\bibitem[{Bock(1985)}]{Bock1985}
\textsc{Bock, M.~E.} (1985).
\newblock Minimax estimators that shift towards a hypersphere for location
  vectors of spherically symmetric distributions.
\newblock \textit{J. Multivariate Anal.} \textbf{17} 127--147.

\bibitem[{Brandwein(1979)}]{Brandwein1979}
\textsc{Brandwein, A.~C.} (1979).
\newblock Minimax estimation of the mean of spherically symmetric distributions
  under general quadratic loss.
\newblock \textit{J. Multivariate Anal.} \textbf{9} 579--588.

\bibitem[{Brandwein and Strawderman(1978)}]{Brand-Straw1978}
\textsc{Brandwein, A.~C.} and \textsc{Strawderman, W.~E.} (1978).
\newblock Minimax estimation of location parameters for spherically symmetric
  unimodal distributions under quadratic loss.
\newblock \textit{Ann. Statist.} \textbf{6} 377--416.

\bibitem[{Brandwein and Strawderman(1991)}]{Brand-Straw1991b}
\textsc{Brandwein, A.~C.} and \textsc{Strawderman, W.~E.} (1991).
\newblock Generalizations of {J}ames-{S}tein estimators under spherical
  symmetry.
\newblock \textit{Ann. Statist.} \textbf{19} 1639--1650.

\bibitem[{Brown(1966)}]{Brown1966}
\textsc{Brown, L.~D.} (1966).
\newblock On the admissibility of invariant estimators of one or more location
  parameters.
\newblock \textit{Ann. Math. Statist} \textbf{37} 1087--1136.

\bibitem[{Brown(1971)}]{Brown1971}
\textsc{Brown, L.~D.} (1971).
\newblock Admissible estimators, recurrent diffusions, and insoluble boundary
  value problems.
\newblock \textit{Ann. Math. Statist.} \textbf{42} 855--903.

\bibitem[{Brown(1979)}]{Brown1979}
\textsc{Brown, L.~D.} (1979).
\newblock A heuristic method for determining admissibility of estimators---with
  applications.
\newblock \textit{Ann. Statist.} \textbf{7} 960--994.

\bibitem[{Brown and Hwang(1982)}]{Brown-Hwang1983}
\textsc{Brown, L.~D.} and \textsc{Hwang, J.~T.} (1982).
\newblock A unified admissibility proof.
\newblock In \textit{Statistical decision theory and related topics, III, Vol.
  1 (West Lafayette, Ind., 1981)}. Academic Press, New York, 205--230.

\bibitem[{Fourdrinier et~al.(1998)Fourdrinier, Strawderman and
  Wells}]{Fourdrinier-etal1998}
\textsc{Fourdrinier, D.}, \textsc{Strawderman, W.~E.} and \textsc{Wells, M.~T.}
  (1998).
\newblock On the construction of {B}ayes minimax estimators.
\newblock \textit{Ann. Statist.} \textbf{26} 660--671.

\bibitem[{Geluk and de~Haan(1987)}]{geluk-dehaan1987}
\textsc{Geluk, J.~L.} and \textsc{de~Haan, L.} (1987).
\newblock \textit{Regular variation, extensions and {T}auberian theorems},
  vol.~40 of \textit{CWI Tract}.
\newblock Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica,
  Amsterdam.

\bibitem[{Gradshteyn and Ryzhik(2000)}]{Gradshteyn-Ryzhik}
\textsc{Gradshteyn, I.~S.} and \textsc{Ryzhik, I.~M.} (2000).
\newblock \textit{Table of integrals, series, and products}.
\newblock Sixth ed. Academic Press Inc., San Diego, CA.
\newblock Translated from the Russian, Translation edited and with a preface by
  Alan Jeffrey and Daniel Zwillinger.

\bibitem[{Maruyama(1998)}]{Maruyama1998}
\textsc{Maruyama, Y.} (1998).
\newblock A unified and broadened class of admissible minimax estimators of a
  multivariate normal mean.
\newblock \textit{J. Multivariate Anal.} \textbf{64} 196--205.

\bibitem[{Maruyama(2003)}]{Maruyama2003}
\textsc{Maruyama, Y.} (2003).
\newblock Admissible minimax estimators of a mean vector of scale mixtures of
  multivariate normal distributions.
\newblock \textit{J. Multivariate Anal.} \textbf{84} 274--283.

\bibitem[{Maruyama(2004)}]{Maruyama2004}
\textsc{Maruyama, Y.} (2004).
\newblock Stein's idea and minimax admissible estimation of a multivariate
  normal mean.
\newblock \textit{J. Multivariate Anal.} \textbf{88} 320--334.

\bibitem[{Maruyama and Takemura(2005)}]{Maruyama-Takemura2005}
\textsc{Maruyama, Y.} and \textsc{Takemura, A.} (2005).
\newblock Admissibility and minimaxity of generalized bayes estimators for
  spherically symmetric family.
\newblock Tech. rep., METR 05-13, The University of Tokyo (available from
  http://www.keisu.t.u-tokyo.ac.jp/Research/METR/2005/METR05-13.pdf).

\bibitem[{Perng(1970)}]{Perng1970}
\textsc{Perng, S.~K.} (1970).
\newblock Inadmissibility of various ``good'' statistical procedures which are
  translation invariant.
\newblock \textit{Ann. Math. Statist.} \textbf{41} 1311--1321.

\bibitem[{Ralescu et~al.(1992)Ralescu, Brandwein and
  Strawderman}]{Ralescu-etal1992}
\textsc{Ralescu, S.}, \textsc{Brandwein, A.~C.} and \textsc{Strawderman, W.~E.}
  (1992).
\newblock Stein estimation for nonnormal spherically symmetric location
  families in three dimensions.
\newblock \textit{J. Multivariate Anal.} \textbf{42} 35--50.

\bibitem[{Stein(1956)}]{Stein1956}
\textsc{Stein, C.} (1956).
\newblock Inadmissibility of the usual estimator for the mean of a multivariate
  normal distribution.
\newblock In \textit{Proceedings of the Third Berkeley Symposium on
  Mathematical Statistics and Probability, 1954--1955, vol. I}. University of
  California Press, Berkeley and Los Angeles.

\bibitem[{Strawderman(1971)}]{Strawderman1971}
\textsc{Strawderman, W.~E.} (1971).
\newblock Proper {B}ayes minimax estimators of the multivariate normal mean.
\newblock \textit{Ann. Math. Statist.} \textbf{42} 385--388.

\bibitem[{Strawderman(1974)}]{Strawderman74}
\textsc{Strawderman, W.~E.} (1974).
\newblock Minimax estimation of location parameters for certain spherically
  symmetric distributions.
\newblock \textit{J. Multivariate Anal.} \textbf{4} 255--264.

\end{thebibliography}
