# Minimaxity under half-Cauchy type priors

Yuzo<EMAIL_ADDRESS>[ Takeru
<EMAIL_ADDRESS>[ Kobe University  The University
of Tokyo & RIKEN Center for Brain Science

###### Abstract

This is a follow-up paper of Polson and Scott (2012, Bayesian Analysis), which
claimed that the half-Cauchy prior is a sensible default prior for a scale
parameter in hierarchical models. For estimation of a normal mean vector under
the quadratic loss, they showed that the Bayes estimator with respect to the
half-Cauchy prior seems to be minimax through numerical experiments. In terms
of the shrinkage coefficient, the half-Cauchy prior has a U-shape and can be
interpreted as a continuous spike and slab prior. In this paper, we consider a
general class of priors with U-shapes and theoretically establish sufficient
conditions for the minimaxity of the corresponding (generalized) Bayes
estimators. We also develop an algorithm for posterior sampling and present
numerical results.

62C20,

minimaxity,

shrinkage,

spike and slab prior,

###### keywords:

[class=MSC]

###### keywords:

and

t1supported by JSPS KAKENHI Grant Numbers 19K11852 and 22K11933 t2supported by
JSPS KAKENHI Grant Numbers 21H05205, 22K17865 and JST Moonshot Grant Number
JPMJMS2024

##  1 Introduction

Consider a normal hierarchical model

$\displaystyle y\mid\beta$ $\displaystyle\sim N_{p}(\beta,I_{p}),$ (1.1)
$\displaystyle\beta\mid\kappa$ $\displaystyle\sim
N_{p}(0,\kappa^{-1}(1-\kappa)I_{p}),$ (1.2) $\displaystyle\kappa$
$\displaystyle\sim\pi(\kappa),$ (1.3)

where the hyperparameter $\kappa\in(0,1)$ specifies the shrinkage coefficient
of the Bayes estimator (posterior mean) of $\beta$:

$\displaystyle\hat{\beta}(y)={\rm E}[\beta\mid y]=(1-{\rm E}[\kappa\mid y])y.$
(1.4)

Polson and Scott (2012) claimed that the hyperprior

$\pi(\kappa)\propto\kappa^{-1/2}(1-\kappa)^{-1/2},$ (1.5)

is a sensible default choice. Since it has a U-shape with

$\lim_{\kappa\to 0}\pi(\kappa)=\lim_{\kappa\to 1}\pi(\kappa)=\infty,$ (1.6)

it may be regarded as a continuous spike and slab prior. See Carvalho, Polson
and Scott (2010) for a related discussion in the context of horseshoe priors.
For the parameterization

$\lambda=\sqrt{\frac{1-\kappa}{\kappa}}\in(0,\infty),$ (1.7)

the prior (1.5) is expressed as

$\pi(\lambda)\propto\frac{1}{1+\lambda^{2}}I_{(0,\infty)}(\lambda),$ (1.8)

which is the reason why the prior (1.5) is called the half-Cauchy prior.

Polson and Scott (2012) introduced a class of “hypergeometric inverted-beta
priors” with the density

$\pi(\kappa)\propto\kappa^{a-1}(1-\kappa)^{b-1}(1+c\kappa)^{-1}\exp(d\kappa),$
(1.9)

which is a generalization of the half-Cauchy prior (1.5). For estimation of
$p$-variate normal mean $\beta$ under the quadratic loss
$\|\hat{\beta}-\beta\|^{2}$, they derived expressions for the risk of Bayes
estimators with respect to the prior (1.9). Recall that the usual estimator
$\hat{\beta}=y$ is inadmissible for $p\geq 3$ although it is minimax for any
$p$ (Stein, 1974; Fourdrinier, Strawderman and Wells, 2018). Through numerical
experiments, Polson and Scott (2012) discussed the minimaxity of the Bayes
estimators under (1.9) for $p\geq 3$ and compared them with the James–Stein
estimator.

In this paper, we consider more general (possibly improper) prior

$\pi(\kappa)=\kappa^{a-1}(1-\kappa)^{b-1}h(\kappa),$ (1.10)

where we assume

  1. A.1

[around $\kappa=0$] $a<1$ and $h(\kappa)$ is slowly varying at $\kappa=0$ with

$\lim_{\kappa\to 0}\kappa\frac{h^{\prime}(\kappa)}{h(\kappa)}=0,$ (1.11)

  2. A.2

[around $\kappa=1$] $0<b<1$ and $h(1)<\infty$.

We derive sufficient conditions for the minimaxity of the (generalized) Bayes
estimator with respect to the prior (1.10) and give several examples. By A.1
and A.2, the prior (1.10) has a U-shape with (1.6) and is hence regarded as a
continuous spike and slab prior. More properties of the prior and the
corresponding marginal density are given in Section 2. Fourdrinier,
Strawderman and Wells (1998) showed the minimaxity of (generalized) Bayes
estimators under priors similar to (1.10), but with $b\geq 1$ in A.2.
Furthermore, the first author of this paper, Maruyama (1998), established the
minimaxity for the case $h(\kappa)\equiv 1$,

$\pi(\kappa)=\kappa^{a-1}(1-\kappa)^{b-1}$ (1.12)

with some $b\in(0,1)$. Hence this paper can be also viewed as an extension of
two papers Fourdrinier, Strawderman and Wells (1998) and Maruyama (1998).

The organization of this paper is as follows. In Section 2, we investigate the
properties of the prior (1.10) and the corresponding marginal density. In
Section 3, we derive sufficient conditions for the minimaxity of the
(generalized) Bayes estimator with respect to the prior (1.10) with A.1 and
A.2. In Section 4, we give several examples of the prior (1.10) for which the
(generalized) Bayes estimator is minimax. While the half-Cauchy prior itself
is not included in these examples, we provide a variant of the half-Cauchy
prior that gives a minimax Bayes estimator. In Section 5, we develop an
algorithm for sampling from the posterior distribution under the prior (1.10)
and present some numerical results.

##  2 Properties of the prior and the marginal density

For the hierarchical model (1.1), (1.2) and (1.3), the prior on $\beta$ is
given by

$\pi(\beta)=\int_{0}^{1}\frac{1}{(2\pi)^{p/2}}\left(\frac{\kappa}{1-\kappa}\right)^{p/2}\exp\left(-\frac{\kappa}{1-\kappa}\frac{\|\beta\|^{2}}{2}\right)\pi(\kappa)\mathrm{d}\kappa,$
(2.1)

and the marginal density of $y$ is given by

$m(y)=\int\frac{1}{(2\pi)^{p/2}}\exp\left(-\frac{\|y-\beta\|^{2}}{2}\right)\pi(\beta)\mathrm{d}\beta.$
(2.2)

The following result summarizes the properties of $\pi(\kappa)$ in (1.10)
satisfying A.1 and A.2 and $m(y)$.

######  Lemma 2.1.

  1. 1.

The prior $\pi(\kappa)$ in (1.10) is proper if either $a>0$ or {$a=0$ and
$\int_{0}^{1}\kappa^{-1}h(\kappa)\mathrm{d}\kappa<\infty$}.

  2. 2.

The marginal density of $y$ with respect to (1.10) is

$m(y)=\frac{1}{(2\pi)^{p/2}}\int_{0}^{1}\exp\left(-\kappa\frac{\|y\|^{2}}{2}\right)\kappa^{p/2+a-1}(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa,$
(2.3)

which is finite for every $y\in\mathbb{R}^{p}$ if either $a>-p/2$ or {$a=-p/2$
and $\int_{0}^{1}\kappa^{-1}h(\kappa)\mathrm{d}\kappa<\infty$}.

###### Proof.

[Part 1] By A.2, $\pi(\kappa)$ is integrable around $\kappa=1$, since

$\lim_{\kappa\to 1}\frac{\pi(\kappa)}{h(1)(1-\kappa)^{b-1}}=1.$ (2.4)

By A.1, $\pi(\kappa)$ satisfies

$\lim_{\kappa\to 0}\frac{\pi(\kappa)}{\kappa^{a-1}h(\kappa)}=1,$ (2.5)

and $\kappa^{a-1}h(\kappa)$ is integrable around $\kappa=0$ if either $a>0$ or
{$a=0$ and $\int_{0}^{1}\kappa^{-1}h(\kappa)\mathrm{d}\kappa<\infty$}.

[Part 2] By the identity

$\|y-\beta\|^{2}+\frac{\kappa}{1-\kappa}\|\beta\|^{2}=\frac{1}{1-\kappa}\left\|\beta-(1-\kappa)y\right\|^{2}+\kappa\|y\|^{2},$
(2.6)

the marginal density is

$\displaystyle m(y)$
$\displaystyle=\int\frac{1}{(2\pi)^{p/2}}\exp\left(-\frac{\|y-\beta\|^{2}}{2}\right)\pi(\beta)\mathrm{d}\beta$
(2.7)
$\displaystyle=\iint\frac{1}{(2\pi)^{p/2}}\exp\left(-\frac{\|y-\beta\|^{2}}{2}\right)\frac{1}{(2\pi)^{p/2}}\left(\frac{\kappa}{1-\kappa}\right)^{p/2}$
$\displaystyle\quad\times\exp\left(-\frac{\kappa}{1-\kappa}\frac{\|\beta\|^{2}}{2}\right)\pi(\kappa)\mathrm{d}\beta\mathrm{d}\kappa$
$\displaystyle=\frac{1}{(2\pi)^{p/2}}\int\exp\left(-\kappa\frac{\|y\|^{2}}{2}\right)\kappa^{p/2}\pi(\kappa)\mathrm{d}\kappa$
$\displaystyle=\frac{1}{(2\pi)^{p/2}}\int_{0}^{1}\exp\left(-\kappa\frac{\|y\|^{2}}{2}\right)\kappa^{p/2+a-1}(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa.$

From $\exp(-\kappa\|y\|^{2}/2)\leq 1$,

$m(y)\leq\frac{1}{(2\pi)^{p/2}}\int_{0}^{1}\kappa^{p/2+a-1}(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa.$
(2.8)

The right hand side of (2.8) is finite (integrable) if either $a>-p/2$ or
{$a=-p/2$ and $\int_{0}^{1}\kappa^{-1}h(\kappa)\mathrm{d}\kappa<\infty$}. ∎

By Tweedie’s formula (Efron, 2011, 2023), the Bayes estimator can be expressed
as

$\hat{\beta}=y+\nabla\log
m(y)=\left(1-\frac{\int_{0}^{1}\kappa^{p/2+1}\exp(-\kappa\|y\|^{2}/2)\pi(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa^{p/2}\exp(-\kappa\|y\|^{2}/2)\pi(\kappa)\mathrm{d}\kappa}\right)y.$
(2.9)

By Lemma 2.1, the marginal density can be finite even when $\pi(\kappa)$ is
improper. In such case, the generalized Bayes estimator is still given by
(2.9). In the next section, we investigate the minimaxity of both proper Bayes
and generalized Bayes estimators.

##  3 Minimaxity

Let

$H(\kappa)=\kappa\frac{h^{\prime}(\kappa)}{h(\kappa)},\quad
H_{1}(\kappa)=\inf_{t\leq\kappa}H(t)\ \text{ and }\
H_{2}(\kappa)=H(\kappa)-H_{1}(\kappa).$ (3.1)

The definition of $H_{1}(\kappa)$ above and (1.11) in A.1 imply that
$H_{1}(\kappa)$ is non-positive and monotone non-increasing. Also,
$H_{2}(\kappa)$ is non-negative and $H_{1}(0)=H_{2}(0)=0$ by (1.11) and (3.1).
Then, we have the following result.

######  Theorem 3.1.

Suppose $-p/2\leq a<p/2-2$. The (generalized) Bayes estimator with respect to
the prior (1.10) is minimax if

$\frac{3p}{2}+a-\frac{p+2a+2+2\max_{\kappa\in[0,1]}H_{2}(\kappa)}{b}+\min\left\\{0,p/2+a+2+H_{1}(1)\right\\}\geq
0.$ (3.2)

In the proof, we utilize the following “correlation inequality” several times.

######  Lemma 3.1.

Suppose $f(x)$ and $g(x)$ are both monotone non-decreasing in $x$. Let $X$ be
a continuous random variable. Then,

$E[f(X)g(X)]\geq E[f(X)]E[g(X)].$ (3.3)

###### Proof of Theorem 3.1.

Since the marginal density $m(y)$ given by (2.3) is spherically symmetric, let
$m_{*}(\|y\|^{2})\coloneqq m(y)$. Then, the quadratic risk of the
(generalized) Bayes estimator (2.9) is

$\begin{split}&E\left[\|\hat{\beta}-\beta\|^{2}\right]=E\left[\left\|Y-2\frac{m^{\prime}_{*}(\|Y\|^{2})}{m_{*}(\|Y\|^{2})}Y-\beta\right\|^{2}\right]\\\
&=E\left[\|Y-\beta\|^{2}+4\left(\frac{m^{\prime}_{*}(\|Y\|^{2})}{m_{*}(\|Y\|^{2})}\right)^{2}\|Y\|^{2}-4\sum_{i=1}^{p}(Y_{i}-\beta_{i})Y_{i}\frac{m^{\prime}_{*}(\|Y\|^{2})}{m_{*}(\|Y\|^{2})}\right]\\\
&=p+4E\left[\frac{m^{\prime}_{*}(\|Y\|^{2})}{m_{*}(\|Y\|^{2})}\Bigl{(}p-2\|Y\|^{2}\frac{m^{\prime\prime}_{*}(\|Y\|^{2})}{-m^{\prime}_{*}(\|Y\|^{2})}+\|Y\|^{2}\frac{-m^{\prime}_{*}(\|Y\|^{2})}{m_{*}(\|Y\|^{2})}\Bigr{)}\right],\end{split}$
(3.4)

where the third equality follows from Stein (1974) identity. Note
$m^{\prime}_{*}(\|y\|^{2})\leq 0$. Thus, a sufficient condition for minimaxity
is

$\begin{split}&p-\|y\|^{2}\frac{\int_{0}^{1}\kappa^{p/2+a+1}(1-\kappa)^{b-1}h(\kappa)\exp(-\kappa\|y\|^{2}/2)\mathrm{d}\kappa}{\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b-1}h(\kappa)\exp(-\kappa\|y\|^{2}/2)\mathrm{d}\kappa}\\\
&\qquad+\frac{\|y\|^{2}}{2}\frac{\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b-1}h(\kappa)\exp(-\kappa\|y\|^{2}/2)\mathrm{d}\kappa}{\int_{0}^{1}\kappa^{p/2+a-1}(1-\kappa)^{b-1}h(\kappa)\exp(-\kappa\|y\|^{2}/2)\mathrm{d}\kappa}\geq
0.\end{split}$ (3.5)

For $w=\|y\|^{2}/2$, the sufficient condition (3.5) is equivalent to

$\Delta(w)\geq 0,$ (3.6)

where

$\begin{split}\Delta(w)&=p-2w\frac{\int_{0}^{1}\kappa^{p/2+a+1}(1-\kappa)^{b-1}h(\kappa)\exp(w\\{1-\kappa\\})\mathrm{d}\kappa}{\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b-1}h(\kappa)\exp(w\\{1-\kappa\\})\mathrm{d}\kappa}\\\
&\quad+w\frac{\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b-1}h(\kappa)\exp(w\\{1-\kappa\\})\mathrm{d}\kappa}{\int_{0}^{1}\kappa^{p/2+a-1}(1-\kappa)^{b-1}h(\kappa)\exp(w\\{1-\kappa\\})\mathrm{d}\kappa}.\end{split}$
(3.7)

Note

$\frac{\mathrm{d}}{\mathrm{d}\kappa}\left\\{-\exp(w\\{1-\kappa\\})+1\right\\}=w\exp(w\\{1-\kappa\\})$
(3.8)

and

$\left[\kappa^{p/2+a+1}(1-\kappa)^{b-1}h(\kappa)\left\\{-\exp(w\\{1-\kappa\\})+1\right\\}\right]_{0}^{1}=0.$
(3.9)

Then we have

$\begin{split}&w\int_{0}^{1}\kappa^{p/2+a+1}(1-\kappa)^{b-1}h(\kappa)\exp(w\\{1-\kappa\\})\mathrm{d}\kappa\\\
&=(p/2+a+1)\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b-1}h(\kappa)\left\\{\exp(w\\{1-\kappa\\})-1\right\\}\mathrm{d}\kappa\\\
&\quad+\int_{0}^{1}\kappa^{p/2+a+1}(1-\kappa)^{b-1}h^{\prime}(\kappa)\left\\{\exp(w\\{1-\kappa\\})-1\right\\}\mathrm{d}\kappa\\\
&\quad-(b-1)\int_{0}^{1}\kappa^{p/2+a+1}(1-\kappa)^{b-2}h(\kappa)\left\\{\exp(w\\{1-\kappa\\})-1\right\\}\mathrm{d}\kappa.\end{split}$
(3.10)

Similarly we have

$\begin{split}&w\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b-1}h(\kappa)\exp(w\\{1-\kappa\\})\mathrm{d}\kappa\\\
&=(p/2+a)\int_{0}^{1}\kappa^{p/2+a-1}(1-\kappa)^{b-1}h(\kappa)\left\\{\exp(w\\{1-\kappa\\})-1\right\\}\mathrm{d}\kappa\\\
&\quad+\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b-1}h^{\prime}(\kappa)\left\\{\exp(w\\{1-\kappa\\})-1\right\\}\mathrm{d}\kappa\\\
&\quad-(b-1)\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b-2}h(\kappa)\left\\{\exp(w\\{1-\kappa\\})-1\right\\}\mathrm{d}\kappa.\end{split}$
(3.11)

Recall $H(\kappa)=\kappa h^{\prime}(\kappa)/h(\kappa)$ and let

$G(\kappa)=\kappa^{p/2+a-1}(1-\kappa)^{b-1}h(\kappa).$ (3.12)

Then, by (3.10) and (3.11), we have

$\Delta(w)=\frac{p}{2}-a-2+(1-b)\Delta_{1}(w)+\Delta_{2}(w),$ (3.13)

where

$\begin{split}\Delta_{1}(w)&=\frac{\int_{0}^{1}\kappa(1-\kappa)^{-1}\left\\{\exp(w\\{1-\kappa\\})-1\right\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}\\\
&\quad-2\frac{\int_{0}^{1}\kappa^{2}(1-\kappa)^{-1}\left\\{\exp(w\\{1-\kappa\\})-1\right\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa},\end{split}$
(3.14) $\displaystyle\Delta_{2}(w)$
$\displaystyle=2\frac{(p/2+a+1)\int_{0}^{1}\kappa
G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}-\frac{(p/2+a)\int_{0}^{1}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}$
(3.15)
$\displaystyle\quad+\frac{\int_{0}^{1}H(\kappa)\\{\exp(w\\{1-\kappa\\})-1\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}-2\frac{\int_{0}^{1}\kappa
H(\kappa)\\{\exp(w\\{1-\kappa\\})-1\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}.$

[The lower bound of $\Delta_{1}(w)$] Note the expansion,

$\exp(w\\{1-\kappa\\})-1=\sum_{j=1}^{\infty}\frac{w^{j}(1-\kappa)^{j}}{j!}.$
(3.16)

Then we have

$\Delta_{1}(w)\geq-2\frac{\int_{0}^{1}\kappa^{2}(1-\kappa)^{-1}\left\\{\exp(w\\{1-\kappa\\})-1\right\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\left\\{\exp(w\\{1-\kappa\\})-1\right\\}G(\kappa)\mathrm{d}\kappa}.$
(3.17)

Since the correlation inequality (Lemma 3.1) gives

$\int_{0}^{1}\frac{\kappa^{2}}{(1-\kappa)}(1-\kappa)^{j}G(\kappa)\mathrm{d}\kappa\leq\frac{\int_{0}^{1}\kappa^{2}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa(1-\kappa)G(\kappa)\mathrm{d}\kappa}\int_{0}^{1}\kappa(1-\kappa)^{j}G(\kappa)\mathrm{d}\kappa$
(3.18)

for $j\geq 1$, we have

$\frac{\int_{0}^{1}\kappa^{2}(1-\kappa)^{-1}\left\\{\exp(w\\{1-\kappa\\})-1\right\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\left\\{\exp(w\\{1-\kappa\\})-1\right\\}G(\kappa)\mathrm{d}\kappa}\leq\frac{\int_{0}^{1}\kappa^{2}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa(1-\kappa)G(\kappa)\mathrm{d}\kappa}$
(3.19)

and

$\Delta_{1}(w)\geq-2\frac{\int_{0}^{1}\kappa^{2}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa(1-\kappa)G(\kappa)\mathrm{d}\kappa}=-2\frac{\int_{0}^{1}\kappa^{p/2+a+1}(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b}h(\kappa)\mathrm{d}\kappa}.$
(3.20)

Further an integration by parts gives

$\begin{split}&b\int_{0}^{1}\kappa^{p/2+a+1}(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa\\\
&=\left[-\kappa^{p/2+a+1}(1-\kappa)^{b}h(\kappa)\right]_{0}^{1}+(p/2+a+1)\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b}h(\kappa)\mathrm{d}\kappa\\\
&\quad+\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b}H(\kappa)h(\kappa)\mathrm{d}\kappa\\\
&\leq\left\\{p/2+a+1+\max_{\kappa\in[0,1]}H_{2}(\kappa)\right\\}\int_{0}^{1}\kappa^{p/2+a}(1-\kappa)^{b}h(\kappa)\mathrm{d}\kappa,\end{split}$
(3.21)

where the inequality follows from

$H(\kappa)=H_{1}(\kappa)+H_{2}(\kappa)\leq
H_{2}(\kappa)\leq\max_{\kappa\in[0,1]}H_{2}(\kappa).$ (3.22)

Then

$\Delta_{1}(w)\geq-2\frac{p/2+a+1+\max_{\kappa\in[0,1]}H_{2}(\kappa)}{b}.$
(3.23)

[The lower bound of $\Delta_{2}(w)$] The correlation inequality (Lemma 3.1)
gives

$\frac{\int_{0}^{1}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}\leq\frac{\int_{0}^{1}\kappa
G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}$
(3.24)

and hence, for the first and second terms of (3.15), we have

$\begin{split}&2\frac{(p/2+a+1)\int_{0}^{1}\kappa
G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}-\frac{(p/2+a)\int_{0}^{1}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}\\\
&\geq\frac{(p/2+a+2)\int_{0}^{1}\kappa
G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}.\end{split}$
(3.25)

By (3.1), we have

$\begin{split}&\frac{\int_{0}^{1}H(\kappa)\\{\exp(w\\{1-\kappa\\})-1\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}-2\frac{\int_{0}^{1}\kappa
H(\kappa)\\{\exp(w\\{1-\kappa\\})-1\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}\\\
&=\sum_{i=1}^{2}\left(\frac{\int_{0}^{1}H_{i}(\kappa)\\{\exp(w\\{1-\kappa\\})-1\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}-2\frac{\int_{0}^{1}\kappa
H_{i}(\kappa)\\{\exp(w\\{1-\kappa\\})-1\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}\right).\end{split}$
(3.26)

Recall $H_{1}(\kappa)$ is monotone non-increasing. Then, by the correlation
inequality (Lemma 3.1), we have

$\frac{\int_{0}^{1}H_{1}(\kappa)\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}\geq\frac{\int_{0}^{1}\kappa
H_{1}(\kappa)\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}$
(3.27)

and hence

$\begin{split}&\frac{\int_{0}^{1}H_{1}(\kappa)\\{\exp(w\\{1-\kappa\\})-1\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}-2\frac{\int_{0}^{1}\kappa
H_{1}(\kappa)\\{\exp(w\\{1-\kappa\\})-1\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}\\\
&\geq\frac{-\int_{0}^{1}\kappa
H_{1}(\kappa)\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa+2\int_{0}^{1}\kappa
H_{1}(\kappa)G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}\\\
&\geq\frac{\int_{0}^{1}\kappa
H_{1}(\kappa)G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}.\end{split}$
(3.28)

Further, for the $H_{2}$ part in (3.26), we have

$\begin{split}&\frac{\int_{0}^{1}H_{2}(\kappa)\\{\exp(w\\{1-\kappa\\})-1\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}-2\frac{\int_{0}^{1}\kappa
H_{2}(\kappa)\\{\exp(w\\{1-\kappa\\})-1\\}G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}\\\
&\geq-2\frac{\int_{0}^{1}\kappa
H_{2}(\kappa)\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}\\\
&\geq-2\max_{\kappa\in[0,1]}H_{2}(\kappa).\end{split}$ (3.29)

Then, by (3.25), (3.28), and (3.29), we have

$\begin{split}\Delta_{2}(w)&\geq\frac{\int_{0}^{1}\\{p/2+a+2+H_{1}(\kappa)\\}\kappa
G(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa\exp(w\\{1-\kappa\\})G(\kappa)\mathrm{d}\kappa}-2\max_{\kappa\in[0,1]}H_{2}(\kappa)\\\
&\geq\min\left\\{0,p/2+a+2+H_{1}(1)\right\\}-2\max_{\kappa\in[0,1]}H_{2}(\kappa).\end{split}$
(3.30)

Recall

$\Delta(w)=\frac{p}{2}-a-2+(1-b)\Delta_{1}(w)+\Delta_{2}(w),$ (3.31)

as in (3.13). Further, as in (3.20) and (3.30), the lower bound of
$\Delta_{1}(w)$ and $\Delta_{2}(w)$ is negative and non-positive respectively.
Hence the sufficient condition $\Delta(w)\geq 0$ cannot hold if $a\geq p/2-2$.
Let $a<p/2-2$. Then, by (3.13), (3.20) and (3.30), we have

$\Delta(w)\geq\frac{3p}{2}+a-\frac{p+2a+2+2\max_{\kappa\in[0,1]}H_{2}(\kappa)}{b}+\min\left\\{0,p/2+a+2+H_{1}(1)\right\\},$
(3.32)

which completes the proof. ∎

When $H(\kappa)$ is monotone non-increasing, it follows that

$H_{1}(\kappa)=H(\kappa)\text{ and }H_{2}(\kappa)\equiv 0$ (3.33)

for (3.1). When $H(\kappa)$ is monotone non-decreasing, it follows that

$H_{1}(\kappa)=0,\ H_{2}(\kappa)=H(\kappa)\text{ and
}\max_{\kappa\in[0,1]}H_{2}(\kappa)=H(1),$ (3.34)

for (3.1). Then we have the following corollary.

######  Corollary 3.1.

Suppose $-p/2\leq a<p/2-2$.

  1. 1.

Suppose $H(\kappa)$ is monotone non-increasing with $H(1)\geq-(p/2+a+2)$. Then
the (generalized) Bayes estimator with respect to the prior (1.10) is minimax
if

$\displaystyle\frac{p+2a+2}{3p/2+a}\leq b<1.$ (3.35)

  2. 2.

Suppose $H(\kappa)$ is monotone non-decreasing with $H(1)<(p/2-a-2)/2$. Then
the (generalized) Bayes estimator with respect to the prior (1.10) is minimax
if

$\frac{p+2a+2+2H(1)}{3p/2+a}\leq b<1.$ (3.36)

Before Fourdrinier, Strawderman and Wells (1998), minimaxity of estimators was
typically shown through so-called Baranchik’s (1970) sufficient condition,
which states that the shrinkage estimator

$\left(1-\frac{\phi(\|y\|^{2})}{\|y\|^{2}}\right)y$ (3.37)

is minimax if $\phi$ is monotone non-decreasing and $0\leq\phi(w)\leq 2(p-2)$.
For the (generalized) Bayes estimator with respect to the prior (1.10), the
function $\phi$ is obtained from Tweedie’s formula (2.9) as

$\phi(\|y\|^{2})=\|y\|^{2}\frac{\int_{0}^{1}\kappa^{p/2+1}\exp(-\kappa\|y\|^{2}/2)\kappa^{a-1}(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa^{p/2}\exp(-\kappa\|y\|^{2}/2)\kappa^{a-1}(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa}.$
(3.38)

Then, we have the following result on the non-monotonicity of $\phi$, which
means that we cannot employ Baranchik’s (1970) sufficient condition.

######  Lemma 3.2.

  1. 1.

$\lim_{\|y\|^{2}\to\infty}\phi(\|y\|^{2})=p+2a$.

  2. 2.

Suppose

$1-b+\liminf_{\kappa\to 0}\frac{h^{\prime}(\kappa)}{h(\kappa)}>0.$ (3.39)

Then $\phi$ is not monotone and approaches $p+2a$ from the above.

Suppose $H(\kappa)=\kappa h^{\prime}(\kappa)/h(\kappa)$ is monotone non-
decreasing. Since $H(0)=0$ by A.1 and (3.1), both $H(\kappa)$ and
$h^{\prime}(\kappa)/h(\kappa)=H(\kappa)/\kappa$ are non-negative for
$\kappa\in[0,1]$. Then we have

$\liminf_{\kappa\to 0}\frac{h^{\prime}(\kappa)}{h(\kappa)}\geq 0\ \text{ and
}\ 1-b+\liminf_{\kappa\to 0}\frac{h^{\prime}(\kappa)}{h(\kappa)}>0.$ (3.40)

Thus, from Part 2 of Lemma 3.2, $\phi$ is non-monotone. Hence, Part 2 of
Corollary 3.1 provides the minimaxity of (generalized) Bayes estimators with
non-monotone $\phi$.

###### Proof of Lemma 3.2.

Let $w=\|y\|^{2}/2$. Then

$\phi(w)=2w\frac{\int_{0}^{1}\kappa^{p/2+a}\exp(-\kappa
w)(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa}{\int_{0}^{1}\kappa^{p/2+a-1}\exp(-\kappa
w)(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa}.$ (3.41)

A Tauberian theorem (see, e.g., Theorem 13.5.4 in Feller (1971)) gives

$\begin{split}\lim_{w\to\infty}\frac{\int_{0}^{1}\kappa^{p/2+a}\exp(-\kappa
w)(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa}{w^{-p/2-a-1}h(1/w)}&=\Gamma(p/2+a+1)\\\
\lim_{w\to\infty}\frac{\int_{0}^{1}\kappa^{p/2+a-1}\exp(-\kappa
w)(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa}{w^{-p/2-a}h(1/w)}&=\Gamma(p/2+a),\end{split}$
(3.42)

which implies that

$\lim_{\|y\|^{2}\to\infty}\phi(\|y\|^{2})=p+2a.$ (3.43)

The way to approach to $p+2a$ determines $(1-\kappa)^{b-1}h(\kappa)$ as
follows. Under the condition (3.39), there exists $0<\kappa_{0}<1$ and
$\epsilon>0$ such that

$\frac{1-b}{1-\kappa}+\frac{h^{\prime}(\kappa)}{h(\kappa)}\geq\epsilon$ (3.44)

for any $\kappa\in[0,\kappa_{0}]$. An integration by parts with (3.8) and
(3.10) gives

$\begin{split}&w\int_{0}^{1}\kappa^{p/2+a}\exp(-\kappa
w)(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa\\\
&=(p/2+a)\int_{0}^{1}\kappa^{p/2+a-1}(1-\kappa)^{b-1}h(\kappa)\left\\{\exp(-\kappa
w)-\exp(-w)\right\\}\mathrm{d}\kappa\\\
&\quad+\int_{0}^{1}\kappa^{p/2+a}\left\\{(1-\kappa)^{b-1}h(\kappa)\right\\}^{\prime}\left\\{\exp(-\kappa
w)-\exp(-w)\right\\}\mathrm{d}\kappa\\\
&=(p/2+a)\int_{0}^{1}\kappa^{p/2+a-1}(1-\kappa)^{b-1}h(\kappa)\exp(-\kappa
w)\mathrm{d}\kappa\\\
&\quad-\frac{(p/2+a)\int_{0}^{1}\kappa^{p/2+a-1}(1-\kappa)^{b-1}h(\kappa)\mathrm{d}\kappa+\int_{0}^{\epsilon_{0}}\kappa^{p/2+a}\left\\{(1-\kappa)^{b-1}h(\kappa)\right\\}^{\prime}\mathrm{d}\kappa}{\exp(w)}\\\
&\quad+\int_{0}^{\epsilon_{0}}\kappa^{p/2+a}\left\\{(1-\kappa)^{b-1}h(\kappa)\right\\}^{\prime}\exp(-\kappa
w)\mathrm{d}\kappa\\\
&\quad+\int_{\epsilon_{0}}^{1}\kappa^{p/2+a}\left\\{(1-\kappa)^{b-1}h(\kappa)\right\\}^{\prime}\left\\{\exp(-\kappa
w)-\exp(-w)\right\\}\mathrm{d}\kappa.\end{split}$ (3.45)

For the last term, we have

$\begin{split}\exp(-\kappa w)-\exp(-w)&=\exp(-\kappa
w)\left\\{1-\exp(-\\{1-\kappa\\}w)\right\\}\\\ &\leq\exp(-\kappa
w)(1-\kappa)w\end{split}$ (3.46)

and hence

$\begin{split}&\left|\int_{\epsilon_{0}}^{1}\kappa^{p/2+a}\left\\{(1-\kappa)^{b-1}h(\kappa)\right\\}^{\prime}\left\\{\exp(-\kappa
w)-\exp(-w)\right\\}\mathrm{d}\kappa\right|\\\
&\leq\frac{w}{\exp(\kappa_{0}w)}\int_{\epsilon_{0}}^{1}\kappa^{p/2+a}\left|\left\\{(1-\kappa)^{b-1}h(\kappa)\right\\}^{\prime}\right|(1-\kappa)\mathrm{d}\kappa.\end{split}$
(3.47)

For the third term,

$\begin{split}&\int_{0}^{\epsilon_{0}}\kappa^{p/2+a}\left\\{(1-\kappa)^{b-1}h(\kappa)\right\\}^{\prime}\exp(-\kappa
w)\mathrm{d}\kappa\\\
&=\int_{0}^{\epsilon_{0}}\kappa^{p/2+a}\left\\{\frac{1-b}{1-\kappa}+\frac{h^{\prime}(\kappa)}{h(\kappa)}\right\\}(1-\kappa)^{b-1}h(\kappa)\exp(-\kappa
w)\mathrm{d}\kappa\\\
&\geq\epsilon\int_{0}^{\epsilon_{0}}\kappa^{p/2+a}(1-\kappa)^{b-1}h(\kappa)\exp(-\kappa
w)\mathrm{d}\kappa.\end{split}$ (3.48)

Thus, by (3.44), we have

$\lim_{w\to\infty}\frac{\int_{0}^{\epsilon_{0}}\kappa^{p/2+a}\left\\{(1-\kappa)^{b-1}h(\kappa)\right\\}^{\prime}\exp(-\kappa
w)\mathrm{d}\kappa}{w^{-p/2-a-1}h(1/w)}\geq\epsilon\Gamma(p/2+a+1)$ (3.49)

and

$\liminf_{\|y\|^{2}\to\infty}\|y\|^{2}\left\\{\phi(\|y\|^{2})-(p+2a)\right\\}>0,$
(3.50)

which implies that $\phi$ is not monotone and approaches $p+2a$ from the
above. ∎

##  4 Choices of $a$, $b$ and $h(\kappa)$

In this section, some interesting choices of $a$, $b$ and $h(\kappa)$ in the
prior (1.10) satisfying Corollary 3.1 are provided.

###  4.1 A variant of half-Cauchy with $h(\kappa)\equiv 1$

Recall that the prior (1.10) coincides with the half-Cauchy prior when
$a=b=1/2$ and $h(\kappa)\equiv 1$. Although Figure 2 of Polson and Scott
(2012) suggests the minimaxity of the Bayes estimator under the half-Cauchy
prior numerically, the choice $a=b=1/2$ is not included in Corollary 3.1 with
$H(\kappa)\equiv 0$. Among the prior (1.10) with $a=b$ and $H(\kappa)\equiv
0$,

$a=b=a_{*}=\frac{-3p+4+\sqrt{9p^{2}-8p+48}}{4}\in(0,1),$ (4.1)

is the smallest choice satisfying Corollary 3.1 for $p\geq 7$. Thus we find
the choice

$a=b=a_{*},\quad h(\kappa)\equiv 1.$ (4.2)

###  4.2 Log-adjusted prior

Let

$h(\kappa)=\left\\{1+c_{1}\log(1/\kappa)\right\\}^{c_{2}},$ (4.3)

for $c_{1}>0$. Then

$H(\kappa)=\kappa\frac{h^{\prime}(\kappa)}{h(\kappa)}=-c_{2}\frac{c_{1}}{1+c_{1}\log(1/\kappa)},$
(4.4)

which is monotone increasing/decreasing if $c_{2}$ is negative/positive,
respectively. By Corollary 3.1 with $H(1)=-c_{1}c_{2}$, we have the following
result.

######  Corollary 4.1.

Let $-p/2\leq a<p/2-2$ and
$h(\kappa)=\left\\{1+c_{1}\log(1/\kappa)\right\\}^{c_{2}}$ with $c_{1}>0$.

  1. 1.

Suppose $0<c_{2}\leq(p/2+a+2)/c_{1}$. Then the (generalized) Bayes estimator
with respect to the prior (1.10) is minimax if

$\frac{p+2a+2}{3p/2+a}\leq b<1.$ (4.5)

  2. 2.

Suppose $-(p/2-a-2)/(2c_{1})<c_{2}<0$. Then the (generalized) Bayes estimator
with respect to the prior (1.10) is minimax if

$\frac{p+2a+2-2c_{1}c_{2}}{3p/2+a}\leq b<1.$ (4.6)

As in Part 1 of Lemma 2.1, the choice

$a=0,\quad c_{2}=-2$ (4.7)

corresponds to a proper prior on the boundary between propriety and
impropriety, since

$\int_{0}^{1}\frac{\mathrm{d}\kappa}{\kappa\\{1+c_{1}\log(1/\kappa)\\}^{2}}=\int_{1}^{\infty}\frac{\mathrm{d}t}{(1+c_{1}t)^{2}}=\left[\frac{1}{c_{1}(1+c_{1}t)}\right]_{1}^{\infty}=\frac{1}{c_{1}(1+c_{1})}$
(4.8)

and

$\int_{0}^{1}\frac{\mathrm{d}\kappa}{\kappa\\{1+c_{1}\log(1/\kappa)\\}}=\int_{1}^{\infty}\frac{\mathrm{d}t}{1+c_{1}t}=\left[\frac{\log(1+c_{1}t)}{c_{1}}\right]_{1}^{\infty}=\infty.$
(4.9)

Note that Hamura, Irie and Sugasawa (2020) utilized a similar U-shape prior
with $a=0$ and $c_{1}<-1$ for the analysis of sparse signals. Let

$c_{1}=\frac{p/2-a-2}{8}=\frac{p-4}{16}.$ (4.10)

Then, by Part 2 of Corollary 4.1, the corresponding proper Bayes estimator is
minimax if $p\geq 5$ and

$\frac{5p+4}{6p}\leq b<1.$ (4.11)

Thus we find the choice

$a=0,\quad b=\frac{5p+4}{6p},\quad c_{1}=\frac{p-4}{16},\quad c_{2}=-2.$
(4.12)

###  4.3 Hypergeometric inverted-beta prior

Let

$h(\kappa)=\left(1+c_{3}\kappa\right)^{c_{4}}\exp(d\kappa),$ (4.13)

for $c_{3}>0$. It can be viewed as a generalization of the “hypergeometric
inverted-beta prior” by Polson and Scott (2012), which corresponds to the
prior (1.10) with $h(\kappa)=(1+c\kappa)^{-1}\exp(d\kappa)$. Then we have

$H(\kappa)=\kappa\frac{h^{\prime}(\kappa)}{h(\kappa)}=d\kappa+c_{4}\frac{c_{3}\kappa}{1+c_{3}\kappa}$
(4.14)

and

$H^{\prime}(\kappa)=d+c_{4}\frac{c_{3}}{(1+c_{3}\kappa)^{2}}.$ (4.15)

######  Lemma 4.1.

  1. 1.

The function $H(\kappa)$ is monotone non-increasing if

$\begin{cases}d\geq 0\text{ and }c_{4}\leq-(c_{3}+1)^{2}d/c_{3},\\\ d<0\text{
and }c_{4}\leq-d/c_{3}.\end{cases}$ (4.16)

  2. 2.

The function $H(\kappa)$ is monotone non-decreasing if

$\begin{cases}d\geq 0\text{ and }c_{4}\geq-d/c_{3}\\\ d<0\text{ and
}c_{4}>-(c_{3}+1)^{2}d/c_{3}.\end{cases}$ (4.17)

###### Proof.

It is clear that $L(\kappa)$ is monotone non-decreasing/non-increasing for
{$d\geq 0$ and $c_{4}\geq 0$} and {$d\leq 0$ and $c_{4}\leq 0$}, respectively.
Further, for $dc_{4}<0$, we have

$\begin{split}H^{\prime}(\kappa)&=\frac{d}{(1+c_{3}\kappa)^{2}}\left((1+c_{3}\kappa)^{2}-\frac{-c_{3}c_{4}}{d}\right)\\\
&=\frac{1+c_{3}\kappa+\sqrt{-c_{3}c_{4}/d}}{(1+c_{3}\kappa)^{2}/c_{3}}d\left(\lambda-\kappa_{*}\right),\end{split}$
(4.18)

where

$\kappa_{*}=\frac{\sqrt{-c_{3}c_{4}/d}-1}{c_{3}}.$ (4.19)

Then $H^{\prime}(\kappa)\geq 0$ follows either if {$d>0$ and $\kappa_{*}\leq
0$} or if {$d<0$ and $\kappa_{*}\geq 1$}. Similarly $H^{\prime}(\kappa)\leq 0$
follows either if {$d>0$ and $\kappa_{*}\geq 1$} or if {$d<0$ and
$\kappa_{*}\leq 0$}. Thus we complete the proof. ∎

By Lemma 4.1 and Corollary 3.1 with $H(1)=d+c_{3}c_{4}/(1+c_{3})$, we have the
following minimaxity result on the Bayes estimators under the hypergeometric
inverted-beta priors, which was not fully investigated in Polson and Scott
(2012).

######  Corollary 4.2.

Let $-p/2\leq a<p/2-2$ and
$h(\kappa)=\left(1+c_{3}\kappa\right)^{c_{4}}\exp(d\kappa)$ with $c_{3}>0$.

  1. 1.

Suppose

$\begin{cases}d\geq 0\text{ and }c_{4}\leq-(c_{3}+1)^{2}d/c_{3},\\\ d<0\text{
and }c_{4}\leq-d/c_{3},\\\ d+c_{3}c_{4}/(1+c_{3})\geq-(p/2-a-2).\end{cases}$
(4.20)

Then the (generalized) Bayes estimator with respect to the prior (1.10) is
minimax if

$\frac{p+2a+2}{3p/2+a}\leq b<1.$ (4.21)

  2. 2.

Suppose

$\begin{cases}d\geq 0\text{ and }c_{4}\geq-d/c_{3},\\\ d<0\text{ and
}c_{4}>-(c_{3}+1)^{2}d/c_{3},\\\
d+c_{3}c_{4}/(1+c_{3})\leq(p/2-a-2)/2.\end{cases}$ (4.22)

Then the (generalized) Bayes estimator with respect to the prior (1.10) is
minimax if

$\frac{p+2a+2+2\\{d+c_{3}c_{4}/(1+c_{3})\\}}{3p/2+a}\leq b<1.$ (4.23)

##  5 Simulation

###  5.1 Algorithm for posterior sampling

We develop an algorithm for sampling from the posterior $\pi(\beta,\kappa\mid
y)$ corresponding to our prior $\pi(\beta,\kappa)$ in (1.2) and (1.10). When
$h(\kappa)\equiv 1$ (e.g. half-Cauchy prior), the Gibbs sampler can be written
down in closed-form by using the fact that the beta-prime (or Pearson Type VI)
distribution is given by an inverse-Gamma mixture of inverse-Gamma (Polson and
Scott, 2012). However, it seems difficult to derive such a data-augmentation
technique for general $h(\kappa)$. Thus, we employ the Metropolis-within-Gibbs
algorithm (Robert and Casella, 2004) here.

First, the conditional distribution $\pi(\beta\mid\kappa,y)$ is Gaussian with
mean $(1-\kappa)y$ and covariance $(1-\kappa)I_{p}$ and thus easy to sample.
Next, the conditional distribution $\pi(\kappa\mid\beta,y)$ is

$\pi(\kappa\mid\beta,y)\propto\kappa^{a+p/2-1}(1-\kappa)^{b-1}\exp\left(-\frac{\|y\|^{2}}{2}\kappa\right)h(\kappa),\quad
0\leq\kappa\leq 1.$

To sample from it, we use the independent Metropolis–Hastings algorithm
(Robert and Casella, 2004) with the proposal distribution given by ${\rm
Beta}(\tilde{a},\tilde{b})$, where we set $\tilde{a}=\max(a,0.5)$ and
$\tilde{b}=0.5$. In summary, our algorithm is given as follows:

  1. 1.

Set $\kappa_{0}=0.5$ and $t=1$

  2. 2.

Sample $\beta_{t}\sim{\rm N}_{p}((1-\kappa_{t-1})y,(1-\kappa_{t-1})I_{p})$

  3. 3.

Sample $\tilde{\kappa}\sim{\rm Beta}(\tilde{a},\tilde{b})$ and set

$\kappa_{t}=\begin{cases}\tilde{\kappa}&{\rm with\ probability}\ q\\\
\kappa_{t-1}&{\rm otherwise}\end{cases}$

where

$q=\min\left(\left(\frac{\tilde{\kappa}}{\kappa}\right)^{a-\tilde{a}+p/2}\left(\frac{1-\tilde{\kappa}}{1-\kappa}\right)^{b-\tilde{b}}\exp\left(-\frac{\|y\|^{2}}{2}(\tilde{\kappa}-\kappa)\right)\frac{h(\tilde{\kappa})}{h(\kappa)},1\right)$

  4. 4.

Update $t$ to $t+1$ and return to 2

In computing the generalized Bayes estimator (posterior mean of $\beta$), we
applied the Rao–Blackwellization technique for variance reduction (Robert and
Roberts, 2021). Namely, we replaced step 2 of the above algorithm with
$\beta_{t}=(1-\kappa_{t-1})y$ and took average of sampled $\beta$ after burn-
in.

In the following, we examine two priors. The first one is

$\displaystyle\pi(\kappa)=\kappa^{a_{*}-1}(1-\kappa)^{a_{*}-1}$ (5.1)

where $a_{*}\in(0,1]$ is given in (4.1). From Corollary 3.1, the Bayes
estimator with respect to this prior is minimax when $p\geq 7$. The second one
is

$\displaystyle\pi(\kappa)=\kappa^{a-1}(1-\kappa)^{b-1}(1+c_{1}\log(1/\kappa))^{c_{2}},$
(5.2)

where $a=0,b=(5p+4)/(6p),c_{1}=(p-4)/16,c_{2}=-2$ as in (4.12). From the
discussion in Section 4.2, this prior is proper and the Bayes estimator with
respect to it is minimax when $p\geq 5$.

###  5.2 Risk comparison

Figure 1 plots the quadratic risk of the Bayes estimators with respect to the
priors (5.1) and (5.2) for $p=10$. The risk of the James–Stein estimator is
also plotted for comparison. For all estimators, the risk is increasing with
$\|\beta\|$ and converges to $p=10$ as $\|\beta\|\to\infty$, which indicates
their minimaxity. The two Bayes estimators attain smaller risk than the
James–Stein estimator when $\|\beta\|\leq 4$, which imply their advantage in
sparse settings.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$2$$4$$6$$8$$10$$\|\beta\|$quadratic risk
Figure 1: Comparison of quadratic risk for $p=10$. solid: Bayes estimator with
respect to the prior (5.1), dashed: Bayes estimator with respect to the prior
(5.2), dotted: James–Stein estimator.

###  5.3 Shrinkage factor

Figure 2 plots the shrinkage factors $\phi(\|y\|^{2})$ in (3.38) of Bayes
estimators with respect to the priors (5.1) and (5.2) for $p=10$. From Lemma
3.2, $\phi(\|y\|^{2})$ for the prior (5.1) is non-monotone and converges to
$p+2a_{*}\approx 11.7309$ from the above. Similarly, $\phi(\|y\|^{2})$ for the
prior (5.2) is non-monotone and converges to $p=10$ from the above. The figure
is compatible with these results and indicates that our sampling algorithm
converged properly.

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$0$$2$$4$$6$$8$$10$$12$$14$$\|y\|$$\phi(\|y\|^{2})$
Figure 2: Comparison of shrinkage factors for $p=10$. solid: Bayes estimator
with respect to prior (5.1), dashed: Bayes estimator with respect to prior
(5.2). dotted: James–Stein estimator.

###  5.4 Prior densities

Figure 3 plots the log-prior densities of $\kappa$. The prior (5.1) shows a
symmetric U-shape with the same divergence speed at $\kappa=0$ and $\kappa=1$.
On the other hand, the prior (5.2) has faster divergence at $\kappa=0$ than at
$\kappa=1$. Note that Carvalho, Polson and Scott (2010) used a similar plot to
compare several types of shrinkage priors.

$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$0$$0.5$$1$$1.5$$2$$2.5$$3$$\kappa$$\log\pi(\kappa)$
Figure 3: Comparison of log-prior densities $\log\pi(\kappa)$ for $p=10$.
solid: prior (5.1), dashed: prior (5.2)

## References

  * Baranchik (1970) [author] Baranchik, A. J.A. J. (1970). A family of minimax estimators of the mean of a multivariate normal distribution. Ann. Math. Statist. 41 642–645. 0253461
  * Carvalho, Polson and Scott (2010) [author] Carvalho, Carlos M.C. M., Polson, Nicholas G.N. G. and Scott, James G.J. G. (2010). The horseshoe estimator for sparse signals. Biometrika 97 465–480. 2650751
  * Efron (2011) [author] Efron, BradleyB. (2011). Tweedie’s formula and selection bias. J. Amer. Statist. Assoc. 106 1602–1614. 10.1198/jasa.2011.tm11181 2896860
  * Efron (2023) [author] Efron, BradleyB. (2023). Machine learning and the JamesâStein estimator. Jpn. J. Stat. Data Sci. in press.
  * Feller (1971) [author] Feller, WilliamW. (1971). An introduction to probability theory and its applications. Vol. II. Second edition. John Wiley & Sons, Inc., New York-London-Sydney. 0270403
  * Fourdrinier, Strawderman and Wells (1998) [author] Fourdrinier, DominiqueD., Strawderman, William E.W. E. and Wells, Martin T.M. T. (1998). On the construction of Bayes minimax estimators. Ann. Statist. 26 660–671. 1626063
  * Fourdrinier, Strawderman and Wells (2018) [author] Fourdrinier, DominiqueD., Strawderman, William E.W. E. and Wells, Martin T.M. T. (2018). Shrinkage estimation. Springer Series in Statistics. Springer, Cham. 3887633
  * Hamura, Irie and Sugasawa (2020) [author] Hamura, YasuyukiY., Irie, KaoruK. and Sugasawa, ShonosukeS. (2020). Shrinkage with Robustness: Log-Adjusted Priors for Sparse Signals. eprint arXiv:2001.08465.
  * Maruyama (1998) [author] Maruyama, YuzoY. (1998). A unified and broadened class of admissible minimax estimators of a multivariate normal mean. J. Multivariate Anal. 64 196–205. 1621863
  * Polson and Scott (2012) [author] Polson, Nicholas G.N. G. and Scott, James G.J. G. (2012). On the half-Cauchy prior for a global scale parameter. Bayesian Anal. 7 887–902. 3000018
  * Robert and Casella (2004) [author] Robert, Christian P.C. P. and Casella, GeorgeG. (2004). Monte Carlo statistical methods, second ed. Springer Texts in Statistics. Springer-Verlag, New York. 10.1007/978-1-4757-4145-2 2080278
  * Robert and Roberts (2021) [author] Robert, Christian P.C. P. and Roberts, GarethG. (2021). Rao-Blackwellisation in the Markov chain Monte Carlo era. Int. Stat. Rev. 89 237–249. 4411904
  * Stein (1974) [author] Stein, CharlesC. (1974). Estimation of the mean of a multivariate normal distribution. In Proceedings of the Prague Symposium on Asymptotic Statistics (Charles Univ., Prague, 1973), Vol. II 345–381. Charles Univ., Prague. 0381062