A class of admissible estimators of multiple regression coefficient with an unknown variance

Abstract

Suppose that we observe $\mathit{y}\mid \mathit{\theta },\tau \sim N_p(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p)$, where $\mathit{\theta }$ is an unknown vector with unknown precision τ. Estimating the regression coefficient $\mathit{\theta }$ with known τ has been well studied. However, statistical properties such as admissibility in estimating $\mathit{\theta }$ with unknown τ are not well studied. Han [(2009). Topics in shrinkage estimation and in causal inference (PhD thesis). Warton School, University of Pennsylvania] appears to be the first to consider the problem, developing sufficient conditions for the admissibility of estimating means of multivariate normal distributions with unknown variance. We generalise the sufficient conditions for admissibility and apply these results to the normal linear regression model. 2-level and 3-level hierarchical models with unknown precision τ are investigated when a standard class of hierarchical priors leads to admissible estimators of $\mathit{\theta }$ under the normalised squared error loss. One reason to consider this problem is the importance of admissibility in the hierarchical prior selection, and we expect that our study could be helpful in providing some reference for choosing hierarchical priors.

Keywords:

• Admissible estimators
• unknown variance
• multivariate normal distributions
• hierarchical models

1. Introduction

Consider a multivariate normal model, (1) $\begin{array}{rl}\mathit{y}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),\text{independently},\\ w\mid \tau & \sim {\chi }_m^2/\tau ,\end{array}$(1) where $\mathit{y}$ is a $p\times 1$ observation vector, $\mathit{\theta }$ is a p-dimensional vector of unknown parameters, and $\tau >0$ is the unknown precision. Statistical properties such as admissibility for estimating $\mathit{\theta }$ can be dated back to James-Stein (1961) when the error variance is known, while the admissibility of generalisation of James-Stein estimator of $\mathit{\theta }$ with unknown parameter τ was studied in Judge, Yancey, and Bock (1983), Fraisse, Raoult, Robert, and Roy (1990), Robert (2007) and so on. For estimating $\mathit{\theta }$ with the unknown nuisance parameter τ in the model (1(1) $\begin{array}{rl}\mathit{y}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),\text{independently},\\ w\mid \tau & \sim {\chi }_m^2/\tau ,\end{array}$(1) ), some authors, such as Strawderman (1973), Maruyama and Strawderman (2005) and Willing and Zhou (2008) studied the minimaxity of Bayesian estimators of $\mathit{\theta }$ under hierarchical priors. The admissibility of a generalised Bayesian estimator of $\mathit{\theta }$ under a class of noninformative priors was recently studied in Han (2009). With additional independent observation $w\mid \tau \sim {\tau }^{-1}{\chi }_m^2,$ Han (2009) found a set of sufficient conditions for the joint priors of $(\mathit{\theta },\tau )$, so that the generalised Bayesian estimator of $\mathit{\theta }$ is admissible under the squared error loss. In practice, we often need to consider a normal linear regression model, (2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2)

where $\mathit{X}$ is $n\times p$ design matrix with full column rank p, $n>p.$ It is of great interest to study the admissibility in estimating the unknown regression coefficients $\mathit{\theta }$ with unknown τ in the normal linear regression model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ).

Several authors have described admissibility as a powerful tool for selecting satisfactory hierarchical generalised Bayesian priors. For example, Berger, Strawderman, and Tang (2005) pointed out that the ‘use of objective improper priors in hierarchical modelling is of enormous practical importance, yet little is known about which such priors are good or bad. It is important that the prior distribution not be too diffuse, and study of admissibility is the most powerful tool known for detecting an over-diffuse prior’. For known precision or error variance, Brown (1971) provided the necessary and sufficient condition of the admissible Bayes estimators under quadratic loss, based on a Markovian representation of the estimation problem. Recent papers related to the theoretical studies of the admissibility of estimators of $\mathit{\theta }$ can be found in Berger and Strawderman (1996), Berger et al. (2005), Berger, Sun, and Song (2018), and so on.

However, most of the literature focussed on models of which variances are given, yet in practical problems, the precision or variance is often unknown. For the admissibility in the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ), to the best of our knowledge, very few results have been obtained because of the technical difficulty. The fundamental tool for proving admissibility for unknown precision is Blyth's method (Blyth, 1951), which proposed a sufficient admissibility condition, relating admissibility of an estimator with the existence of a sequence of prior distributions approximating this estimator. Based on Blyth's results, Han's (2009) found sufficient conditions for the joint priors of $(\mathit{\theta },\tau )$ for model (1(1) $\begin{array}{rl}\mathit{y}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),\text{independently},\\ w\mid \tau & \sim {\chi }_m^2/\tau ,\end{array}$(1) ). Sometimes, those sufficient conditions are strict and difficult to satisfy. We will generalise the sufficient conditions for admissibility and apply these results to the normal linear regression model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ). Using the generalised conditions, a 2-level and 3-level hierarchical models with unknown precision τ are investigated when a standard class of hierarchical priors leads to admissible estimators of $\mathit{\theta }$ under the normalised squared error loss. One motivation to consider this problem is the importance of admissibility in the hierarchical prior selection, and we expect that our study could be helpful to provide some reference for choosing hierarchical priors.

The paper is organised as follows. In Section 2, we introduce the sufficient conditions for admissibility of the generalised Bayesian estimators of $\mathit{\theta }$ for the model (1(1) $\begin{array}{rl}\mathit{y}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),\text{independently},\\ w\mid \tau & \sim {\chi }_m^2/\tau ,\end{array}$(1) ), which is studied by Han (2009). In Section 3, we generalise the sufficient conditions for admissibility and apply these results to the normal linear regression model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ). 2-level and a 3-level hierarchical models with unknown precision τ are investigated in Sections 4 and 5, determining when a standard class of hierarchical priors leads to admissible estimators of $\mathit{\theta }$ under the normalised squared error loss. Finally, some comments are made in Section 6.

2. Han's (2009) results for (1)

Recall the model (1(1) $\begin{array}{rl}\mathit{y}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),\text{independently},\\ w\mid \tau & \sim {\chi }_m^2/\tau ,\end{array}$(1) ) considered in Han (2009), i.e. $(\mathit{y}\mid \mathit{\theta },\tau )\sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),(w\mid \tau )\sim {\tau }^{-1}{\chi }_m^2,$ where $\mathit{y}=(y_1,\dots ,y_p{)}^{\prime }$ and w are independent of each other. Let $\hat{\mathit{\theta }}\equiv \hat{\mathit{\theta }}(\mathit{y},w)$ denote an estimator of $\mathit{\theta }=({\theta }_1,\dots ,{\theta }_p{)}^{\prime }$. Correspondingly, the squared error loss function of $\hat{\mathit{\theta }}$ becomes (3) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }(\hat{\mathit{\theta }}-\mathit{\theta }).$(3) Han (2009) studied a class of prior density for $(\mathit{\theta },\tau )$ with assumption (4) $\pi (\mathit{\theta },\tau )={\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right).$(4) Consequently, the generalised Bayes estimator for the normal mean $\mathit{\theta }$ is the posterior mean of $\mathit{\theta }$, given by (5) $\begin{array}{r}{\hat{\mathit{\theta }}}_B(\mathit{y},w)=\frac{\begin{array}{c}{\int }_{{\mathbb{R}}^p}{\int }_0^{\mathrm{\infty }}\tau \mathit{\theta }{f}_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ {\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\end{array}}{\begin{array}{c}{\int }_{{\mathbb{R}}^p}{\int }_0^{\mathrm{\infty }}\tau f_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ {\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\end{array}},\end{array}$(5) where $\begin{array}{rl}{f}_1(\mathit{y}\mid \mathit{\theta },\tau )& \propto {\tau }^{p/2}\exp \left(-\frac{\tau }{2}\parallel \mathit{y}-\mathit{\theta }{\parallel }^2\right),\\ f_2(w\mid \tau )& \propto w^{(m-2)/2}{\tau }^{m/2}\exp \left(-\frac{\tau w}{2}\right).\end{array}$ Let $m(\mathit{y},w,\tau )$ be the marginal likelihood function of $(\mathit{y},w,\tau )$ with the form $\begin{array}{rl}m(\mathit{y},w,\tau )& ={\int }_{{\mathbb{R}}^p}{f}_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ & \times {\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\mathit{\theta }.\end{array}$ From Brown (1971), the generalised Bayes estimator in (5(5) $\begin{array}{r}{\hat{\mathit{\theta }}}_B(\mathit{y},w)=\frac{\begin{array}{c}{\int }_{{\mathbb{R}}^p}{\int }_0^{\mathrm{\infty }}\tau \mathit{\theta }{f}_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ {\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\end{array}}{\begin{array}{c}{\int }_{{\mathbb{R}}^p}{\int }_0^{\mathrm{\infty }}\tau f_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ {\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\end{array}},\end{array}$(5) ) can be expressed as (6) ${\hat{\mathit{\theta }}}_B(\mathit{y},w)=\mathit{y}+\frac{{\int }_0^{\mathrm{\infty }}{\mathrm{\nabla }}_{\mathit{y}}m(\mathit{y},w,\tau )\mathrm{d}\tau }{{\int }_0^{\mathrm{\infty }}\tau m(\mathit{y},w,\tau )\mathrm{d}\tau },$(6) where ∇ denotes the gradient. Let S denote the ball of radius 1 at the origin in ${\mathrm{I}\mathrm{R}}^{\mathrm{p}}$ and $S^c$ be the complement of S, defined $a\vee b=max(a,b)$. For the hierarchical Bayes model (1(1) $\begin{array}{rl}\mathit{y}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),\text{independently},\\ w\mid \tau & \sim {\chi }_m^2/\tau ,\end{array}$(1) ), Han (2009) studied the admissible generalised Bayes estimators ${\hat{\mathit{\theta }}}_B(\mathit{y},w)$ under the following sufficient conditions.

• Condition 1. ${\int }_{S^c}{\int }_0^{\mathrm{\infty }}\left(1/\tau \right)({\pi }_0(\mathit{\theta }\mid \tau )/\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2\left)\right){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }<\mathrm{\infty }$;

• Condition 2. ${\int }_{S^c}{\int }_0^{\mathrm{\infty }}{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }<\mathrm{\infty }$;

• Condition 3. ${\int }_{S^c}{\int }_0^{\mathrm{\infty }}\tau \parallel \mathit{\theta }{\parallel }^2{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }<\mathrm{\infty }$;

• Condition 4. ${\int }_{S^c}{\int }_0^{\mathrm{\infty }}\left(1/\tau \right)\left(\parallel {\mathrm{\nabla }}_{\mathit{\theta }}{\pi }_0(\mathit{\theta }\mid \tau ){\parallel }^2/{\pi }_0\right(\mathit{\theta }\mid \tau \left)\right){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }<\mathrm{\infty }$;

• Condition 5. For any positive constant B, ${\int }_{\parallel \mathit{\theta }{\parallel }^2<B}{\int }_{\tau <B}{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }<\mathrm{\infty }$;

• Condition 6. Define two sequences of functions (7) $\begin{array}{rl}{h}_j\left(\mathit{\theta }\right)& =\left\{\begin{array}{ll}1,& \text{if}\parallel \mathit{\theta }\parallel <1;\\ 1-\frac{\log (\parallel \mathit{\theta }\parallel )}{\log j},& 1\le \parallel \mathit{\theta }\parallel \le j;\\ 0,& \parallel \mathit{\theta }\parallel >j,\end{array}\right.\text{and}\\ l_j\left(\tau \right)& =\left\{\begin{array}{ll}1,& \text{if}\tau <1;\\ 1-\frac{\log \left(\tau \right)}{\log j},& 1\le \tau \le j;\\ 0,& \tau >j.\end{array}\right.\end{array}$(7) Write $H_j(\mathit{\theta }\mid \tau )=h_j\left(\mathit{\theta }\right){\pi }_0(\mathit{\theta }\mid \tau )$ and $L_j\left(\tau \right)=l_j\left(\tau \right){\pi }_1\left(\tau \right).$ There is a constant C>0, such that $\begin{array}{rl}& \frac{{\int }_S{\int }_0^{\mathrm{\infty }}\tau f_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )H_j(\mathit{\theta }\mid \tau )L_j\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }}{{\int }_{S^c}{\int }_0^{\mathrm{\infty }}\tau f_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )H_j(\mathit{\theta }\mid \tau )L_j\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }}\\ & <C,\mathrm{\forall }\mathit{y},w.\end{array}$

Theorem 2.1

Han, 2009

Consider the model (1(1) $\begin{array}{rl}\mathit{y}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),\text{independently},\\ w\mid \tau & \sim {\chi }_m^2/\tau ,\end{array}$(1) ) with the prior densities ${\pi }_0(\mathit{\theta }\mid \tau )$ and ${\pi }_1\left(\tau \right)$ satisfying Conditions 1–6. If ${\pi }_0(\mathit{\theta }\mid \tau )$ is decreasing with respect to $\parallel \theta \parallel ,$ the corresponding generalised Bayes estimator (6(6) ${\hat{\mathit{\theta }}}_B(\mathit{y},w)=\mathit{y}+\frac{{\int }_0^{\mathrm{\infty }}{\mathrm{\nabla }}_{\mathit{y}}m(\mathit{y},w,\tau )\mathrm{d}\tau }{{\int }_0^{\mathrm{\infty }}\tau m(\mathit{y},w,\tau )\mathrm{d}\tau },$(6) ) for $\mathit{\theta }$ is admissible under the squared error loss function (3(3) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }(\hat{\mathit{\theta }}-\mathit{\theta }).$(3) ).

3. Main results for (2)

We are primarily interested in the normal linear regression model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ). For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ), we let $\tilde{\mathit{y}}=({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}\mathit{X}\mathit{y}$ denote the least squared estimators of $\mathit{\theta }$, and $w={\mathit{y}}^{\prime }({\mathit{I}}_n-\mathit{X}({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}{\mathit{X}}^{\prime })\mathit{y}$ be the usual residual sum squares errors (SSE). Then (8) $\begin{array}{rl}\tilde{\mathit{y}}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}),\text{and}\\ w\mid \tau & \sim {\tau }^{-1}{\chi }_{n-p}^2,\end{array}$(8) independently. Here we obtain w automatically with m=n−p. For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ), consider the normalised squared error loss function of $\hat{\mathit{\theta }}$ given by (9) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }{\mathit{X}}^{\prime }\mathit{X}(\hat{\mathit{\theta }}-\mathit{\theta }).$(9) The corresponding risk function of $\hat{\mathit{\theta }}$ is (10) $R(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=E_{\mathit{\theta },\tau }L(\mathit{\theta },\tau ;\hat{\mathit{\theta }}),\mathit{\theta }\in {\mathbb{R}}^p.$(10) An estimator ${\hat{\mathit{\theta }}}_1$ is inadmissible if there exists another estimator whose risk function is nowhere bigger and somewhere smaller. If no such better estimator exists, ${\hat{\mathit{\theta }}}_1$ is admissible.

For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ), to obtain the admissible estimator of $\mathit{\theta }$ under the normalised squared error loss (9(9) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }{\mathit{X}}^{\prime }\mathit{X}(\hat{\mathit{\theta }}-\mathit{\theta }).$(9) ), we define (11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) where $\mathit{T}$ is a $p\times p$ matrix such that ${\mathit{T}}^{\prime }({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}\mathit{T}={\mathit{I}}_p$.

Lemma 3.1

For the model (1(1) $\begin{array}{rl}\mathit{y}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),\text{independently},\\ w\mid \tau & \sim {\chi }_m^2/\tau ,\end{array}$(1) ), assume the estimator ${\hat{\mathit{\theta }}}_B(\mathit{y},w)$ in (6(6) ${\hat{\mathit{\theta }}}_B(\mathit{y},w)=\mathit{y}+\frac{{\int }_0^{\mathrm{\infty }}{\mathrm{\nabla }}_{\mathit{y}}m(\mathit{y},w,\tau )\mathrm{d}\tau }{{\int }_0^{\mathrm{\infty }}\tau m(\mathit{y},w,\tau )\mathrm{d}\tau },$(6) ) is admissible under the loss function (3(3) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }(\hat{\mathit{\theta }}-\mathit{\theta }).$(3) ). Then the estimator ${\mathit{\delta }}_B\left(\mathit{y}\right)$ in (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) is admissible under the normalised squared error loss (9(9) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }{\mathit{X}}^{\prime }\mathit{X}(\hat{\mathit{\theta }}-\mathit{\theta }).$(9) ) for the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ).

Proof.

Note that the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ) is equivalent to (8(8) $\begin{array}{rl}\tilde{\mathit{y}}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}),\text{and}\\ w\mid \tau & \sim {\tau }^{-1}{\chi }_{n-p}^2,\end{array}$(8) ). It yields that (12) $({\mathit{T}}^{\prime }\tilde{\mathit{y}}\mid \mathit{\theta },\tau )\sim N_p({\mathit{T}}^{\prime }\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p).$(12) It follows from the admissibility of ${\hat{\mathit{\theta }}}_B(\mathit{y},w)$ under the model (1(1) $\begin{array}{rl}\mathit{y}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),\text{independently},\\ w\mid \tau & \sim {\chi }_m^2/\tau ,\end{array}$(1) ) that the estimator ${\hat{\mathit{\theta }}}_B({\mathit{T}}^{\prime }\tilde{\mathit{y}},w)$ for ${\mathit{T}}^{\prime }\mathit{\theta }$ is admissible under the loss function $\begin{array}{rl}L({\mathit{T}}^{\prime }\mathit{\theta },\tau ;{\hat{\mathit{\theta }}}_B({\mathit{T}}^{\prime }\tilde{\mathit{y}},w\left)\right)& =\tau {\left[{\hat{\mathit{\theta }}}_B({\mathit{T}}^{\prime }\tilde{\mathit{y}},w)-{\mathit{T}}^{\prime }\mathit{\theta }\right]}^{\prime }\\ & \times \left[{\hat{\mathit{\theta }}}_B({\mathit{T}}^{\prime }\tilde{\mathit{y}},w)-{\mathit{T}}^{\prime }\mathit{\theta }\right]\\ & =\tau {\left[\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w)-\mathit{\theta }\right]}^{\prime }\left({\mathit{X}}^{\prime }\mathit{X}\right)\\ & \times \left[\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w)-\mathit{\theta }\right]\\ & =\tau ({\mathit{\delta }}_B-\mathit{\theta }{)}^{\prime }{\mathit{X}}^{\prime }\mathit{X}({\mathit{\delta }}_B-\mathit{\theta }).\end{array}$ The proof of this lemma is completed.

Combining Theorem 1 with Lemma 1, we can reach the following theorem.

Theorem 3.2

For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ) with the prior densities ${\pi }_0(\mathit{\theta }\mid \tau )$ and ${\pi }_1\left(\tau \right)$ satisfying Conditions 1–6, suppose that ${\pi }_0(\mathit{\theta }\mid \tau )$ is decreasing with respect to $\parallel \theta \parallel ,$ then ${\mathit{\delta }}_B\left(\mathit{y}\right)$ defined in (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) is the admissible estimator of $\mathit{\theta }$ under the normalised squared error loss (9(9) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }{\mathit{X}}^{\prime }\mathit{X}(\hat{\mathit{\theta }}-\mathit{\theta }).$(9) ).

Theorem 2.1 applies to the case where ${\pi }_0(\mathit{\theta }\mid \tau )$ is spherically symmetric of $\mathit{\theta }$ and decreases in $\parallel \mathit{\theta }\parallel$. As discussed in Han (2009), this requirement is not unique and can be replaced by the following condition.

Condition 7. Denote (13) $\begin{array}{rl}{\mathit{u}}_1& ={\int }_0^{\mathrm{\infty }}{\int }_S{f}_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau ){\mathrm{\nabla }}_{\mathit{\theta }}\\ & \times {\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\mathit{\theta }\mathrm{d}\tau ,\end{array}$(13) (14) $\begin{array}{rl}{z}_1& ={\int }_0^{\mathrm{\infty }}{\int }_S\tau f_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ & \times {\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\mathit{\theta }\mathrm{d}\tau ,\end{array}$(14) (15) $\begin{array}{rl}{\mathit{u}}_{2j}& ={\int }_0^{\mathrm{\infty }}{\int }_S{f}_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ & \times {\mathrm{\nabla }}_{\mathit{\theta }}{H}_j(\mathit{\theta }\mid \tau )L_j\left(\tau \right)\mathrm{d}\mathit{\theta }\mathrm{d}\tau ,\end{array}$(15) (16) $\begin{array}{rl}{z}_{2j}& ={\int }_0^{\mathrm{\infty }}{\int }_S\tau f_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ & \times H_j(\mathit{\theta }\mid \tau )L_j\left(\tau \right)\mathrm{d}\mathit{\theta }\mathrm{d}\tau .\end{array}$(16) We have (17) $\begin{array}{rl}\frac{\parallel {\mathit{u}}_1\parallel }{z_1}& \le \parallel \mathit{y}\parallel ,\end{array}$(17) (18) $\begin{array}{rl}\frac{\parallel {\mathit{u}}_{2j}\parallel }{z_{2j}}& \le \parallel \mathit{y}\parallel ,\text{for any}j=1,2,\dots .\end{array}$(18) As an immediate corollary, we have the following result.

Theorem 3.3

For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ), assume that the prior densities ${\pi }_0(\mathit{\theta }\mid \tau )$ and ${\pi }_1\left(\tau \right)$ satisfy Conditions 1–7. Then estimator ${\mathit{\delta }}_B\left(\mathit{y}\right)$ in (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) for $\mathit{\theta }$ is admissible under the normalised squared error loss (9(9) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }{\mathit{X}}^{\prime }\mathit{X}(\hat{\mathit{\theta }}-\mathit{\theta }).$(9) ).

It might be difficult to show that the ${\pi }_0(\mathit{\theta }\mid \tau )$ is a decreasing function of $\parallel \mathit{\theta }\parallel$. Interestingly, this requirement can be relaxed to the requirement that ${\pi }_0(\mathit{\theta }\mid \tau )$ is a decreasing function of its component, ${\theta }_i^2$, for $i=1,\dots ,p$.

Lemma 3.4

For the model (1(1) $\begin{array}{rl}\mathit{y}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),\text{independently},\\ w\mid \tau & \sim {\chi }_m^2/\tau ,\end{array}$(1) ) with given τ, ${\pi }_0(\mathit{\theta }\mid \tau )$ is a decreasing function of ${\theta }_i^2,$ for $i=1,\dots ,p,$ then Condition 7 holds.

Proof.

For any given $\mathit{y}$, there is an orthogonal matrix $\mathit{Q}$, such that $\mathit{Q}\mathit{y}=(\parallel \mathit{y}\parallel ,0,\dots ,0{)}^{\prime }.$ Without loss of generality, we can transform the coordinate system of $\mathit{\theta }$ such that $\mathit{y}=(\parallel \mathit{y}\parallel ,0,\dots ,0{)}^{\prime }.$ Then, $\begin{array}{rl}{f}_1(\mathit{y}\mid \mathit{\theta },\tau )& =(2\pi {)}^{-p/2}{\tau }^{p/2}\exp \left(-\tau \frac{\parallel \mathit{y}{\parallel }^2+\parallel \mathit{\theta }{\parallel }^2}{2}\right)\\ & \times \exp \left(\tau {\theta }_1\parallel \mathit{y}\parallel \right).\end{array}$ It is easy to verify that the ith coordinate of ${\mathit{u}}_1$ is (19) $\begin{array}{rl}{v}_{1i}& ={\int }_0^{\mathrm{\infty }}{\int }_S{f}_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ & \times \frac{\mathrm{\partial }{\pi }_0(\mathit{\theta }\mid \tau )}{\mathrm{\partial }{\theta }_i}{\pi }_1\left(\tau \right)\mathrm{d}\mathit{\theta }\mathrm{d}\tau .\end{array}$(19) Since ${\pi }_0(\mathit{\theta }\mid \tau )$ is a function of $({\theta }_1^2,\dots ,{\theta }_p^2)$, $f_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\left(\mathrm{\partial }{\pi }_0(\mathit{\theta }\mid \tau )/\mathrm{\partial }{\theta }_i\right){\pi }_1\left(\tau \right)$ is an odd function for ${\theta }_i$ when $i=2,\dots ,p$. It yields $v_{12}=\cdots =v_{1p}=0$. Therefore, $\parallel {\mathit{u}}_1\parallel =|v_{11}|$ and $\parallel {\mathit{u}}_1\parallel /z_1=|v_{11}|/z_1$.

Let $f(\mathit{y},w,\mathit{\theta },\tau )$ be the joint density of $(\mathit{y},w,\mathit{\theta },\tau )$, i.e. $f(\mathit{y},w,\mathit{\theta },\tau )=f_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau ){\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right).$ Using (19(19) $\begin{array}{rl}{v}_{1i}& ={\int }_0^{\mathrm{\infty }}{\int }_S{f}_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ & \times \frac{\mathrm{\partial }{\pi }_0(\mathit{\theta }\mid \tau )}{\mathrm{\partial }{\theta }_i}{\pi }_1\left(\tau \right)\mathrm{d}\mathit{\theta }\mathrm{d}\tau .\end{array}$(19) ), we get (20) $\begin{array}{rl}& {\int }_0^{\mathrm{\infty }}{\int }_S\frac{\mathrm{\partial }f(\mathit{y},w,\mathit{\theta },\tau )}{\mathrm{\partial }{\theta }_1}\mathrm{d}\mathit{\theta }\mathrm{d}\tau \\ & ={\int }_0^{\mathrm{\infty }}{\int }_S\frac{\mathrm{\partial }{f}_1(\mathit{y}\mid \mathit{\theta },\tau )}{\mathrm{\partial }{\theta }_1}{f}_2(w\mid \tau )\\ & \times {\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\mathit{\theta }\mathrm{d}\tau \\ & +{\int }_0^{\mathrm{\infty }}{\int }_S{f}_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ & \times \frac{\mathrm{\partial }{\pi }_0(\mathit{\theta }\mid \tau )}{\mathrm{\partial }{\theta }_1}{\pi }_1\left(\tau \right)\mathrm{d}\mathit{\theta }\mathrm{d}\tau \\ & ={\int }_0^{\mathrm{\infty }}{\int }_S(\parallel \mathit{y}\parallel -{\theta }_1)\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}\mathit{\theta }\mathrm{d}\tau +v_{11}.\end{array}$(20) By the Divergence Theorem (Katz, 2005), (21) $\begin{array}{rl}& {\int }_0^{\mathrm{\infty }}{\int }_S\frac{\mathrm{\partial }f(\mathit{y},w,\mathit{\theta },\tau )}{\mathrm{\partial }{\theta }_1}\mathrm{d}\mathit{\theta }\mathrm{d}\tau \\ & ={\int }_0^{\mathrm{\infty }}{\int }_{\mathrm{\partial }S}f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\theta }_2\cdots \mathrm{d}{\theta }_p\mathrm{d}\tau ,\end{array}$(21) where $\mathrm{\partial }S$ is the boundary of S. Combining (20(20) $\begin{array}{rl}& {\int }_0^{\mathrm{\infty }}{\int }_S\frac{\mathrm{\partial }f(\mathit{y},w,\mathit{\theta },\tau )}{\mathrm{\partial }{\theta }_1}\mathrm{d}\mathit{\theta }\mathrm{d}\tau \\ & ={\int }_0^{\mathrm{\infty }}{\int }_S\frac{\mathrm{\partial }{f}_1(\mathit{y}\mid \mathit{\theta },\tau )}{\mathrm{\partial }{\theta }_1}{f}_2(w\mid \tau )\\ & \times {\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\mathit{\theta }\mathrm{d}\tau \\ & +{\int }_0^{\mathrm{\infty }}{\int }_S{f}_1(\mathit{y}\mid \mathit{\theta },\tau )f_2(w\mid \tau )\\ & \times \frac{\mathrm{\partial }{\pi }_0(\mathit{\theta }\mid \tau )}{\mathrm{\partial }{\theta }_1}{\pi }_1\left(\tau \right)\mathrm{d}\mathit{\theta }\mathrm{d}\tau \\ & ={\int }_0^{\mathrm{\infty }}{\int }_S(\parallel \mathit{y}\parallel -{\theta }_1)\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}\mathit{\theta }\mathrm{d}\tau +v_{11}.\end{array}$(20) ) and (21(21) $\begin{array}{rl}& {\int }_0^{\mathrm{\infty }}{\int }_S\frac{\mathrm{\partial }f(\mathit{y},w,\mathit{\theta },\tau )}{\mathrm{\partial }{\theta }_1}\mathrm{d}\mathit{\theta }\mathrm{d}\tau \\ & ={\int }_0^{\mathrm{\infty }}{\int }_{\mathrm{\partial }S}f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\theta }_2\cdots \mathrm{d}{\theta }_p\mathrm{d}\tau ,\end{array}$(21) ), we get $\begin{array}{rl}{v}_{11}& ={\int }_0^{\mathrm{\infty }}{\int }_{\mathrm{\partial }S}f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\theta }_2\cdots \mathrm{d}{\theta }_p\mathrm{d}\tau \\ & -{\int }_0^{\mathrm{\infty }}{\int }_S(\parallel \mathit{y}\parallel -{\theta }_1)\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}\mathit{\theta }\mathrm{d}\tau .\end{array}$ Then, we have $\begin{array}{rl}\frac{v_{11}}{z_1}& =\frac{{\int }_0^{\mathrm{\infty }}{\int }_{\mathrm{\partial }S}f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\theta }_2\cdots \mathrm{d}{\theta }_p\mathrm{d}\tau }{{\int }_0^{\mathrm{\infty }}{\int }_S\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\theta }_1\cdots \mathrm{d}{\theta }_p\mathrm{d}\tau }\\ & -\frac{{\int }_0^{\mathrm{\infty }}{\int }_S(\parallel \mathit{y}\parallel -{\theta }_1)\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}\mathit{\theta }\mathrm{d}\tau }{{\int }_0^{\mathrm{\infty }}{\int }_S\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\theta }_1\cdots \mathrm{d}{\theta }_p\mathrm{d}\tau }\\ & =\frac{{\int }_0^{\mathrm{\infty }}{\int }_{\mathrm{\partial }S}f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\theta }_2\cdots \mathrm{d}{\theta }_p\mathrm{d}\tau }{{\int }_0^{\mathrm{\infty }}{\int }_S\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\theta }_1\cdots \mathrm{d}{\theta }_p\mathrm{d}\tau }\\ & -\parallel \mathit{y}\parallel +\frac{{\int }_0^{\mathrm{\infty }}{\int }_S{\theta }_1\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}\mathit{\theta }\mathrm{d}\tau }{{\int }_0^{\mathrm{\infty }}{\int }_S\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\mathit{\theta }}_1\mathrm{d}\tau }.\end{array}$ Clearly, $\frac{{\int }_0^{\mathrm{\infty }}{\int }_{\mathrm{\partial }S}f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\theta }_2\cdots \mathrm{d}{\theta }_p\mathrm{d}\tau }{{\int }_0^{\mathrm{\infty }}{\int }_S\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\theta }_1\cdots \mathrm{d}{\theta }_p\mathrm{d}\tau }\ge 0.$ Since $f(\mathit{y},w,\mathit{\theta },\tau )$ can be written by $\begin{array}{rl}f(\mathit{y},w,\mathit{\theta },\tau )& =(2\pi {)}^{-p/2}{\tau }^{p/2}\exp \left(-\frac{\parallel \mathit{y}{\parallel }^2+\parallel \mathit{\theta }{\parallel }^2}{2}\tau \right)\\ & \times \exp (\parallel \mathit{y}\parallel {\theta }_1\tau )f_2\left(w|\tau \right){\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right),\end{array}$ and ${\pi }_0(\mathit{\theta }\mid \tau )$ is symmetric about ${\theta }_1$, then $\frac{{\int }_0^{\mathrm{\infty }}{\int }_S{\theta }_1\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}\mathit{\theta }\mathrm{d}\tau }{{\int }_0^{\mathrm{\infty }}{\int }_S\tau f(\mathit{y},w,\mathit{\theta },\tau )\mathrm{d}{\mathit{\theta }}_1\mathrm{d}\tau }\ge 0.$ Therefore, $\frac{v_{11}}{z_1}\ge -\parallel \mathit{y}\parallel .$ Since ${\pi }_0(\mathit{\theta }\mid \tau )$ is an even function for ${\theta }_1$ and decreasing in ${\theta }_1^2$, then $v_{11}<0$. Therefore, we have $\parallel {\mathit{u}}_1\parallel /z_1=|v_{11}|/z_1\le \parallel \mathit{y}\parallel$. With the same argument as above, we have $\parallel {\mathit{u}}_2\parallel /z_{2j}\le \parallel \mathit{y}\parallel ,$ for any $j=1,2,\dots$.

Consequently, we obtain the following result.

Theorem 3.5

For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ), assume that the prior densities ${\pi }_0(\mathit{\theta }\mid \tau )$ and ${\pi }_1\left(\tau \right)$ satisfy Conditions 1–6. If for any given $\tau >0,$ ${\pi }_0(\mathit{\theta }\mid \tau )$ is decreasing in ${\theta }_i^2,$ $i=1,\dots ,p$. the estimator ${\mathit{\delta }}_B\left(\mathit{y}\right)$ in (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) for $\mathit{\theta }$ is admissible under the normalised squared error loss (9(9) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }{\mathit{X}}^{\prime }\mathit{X}(\hat{\mathit{\theta }}-\mathit{\theta }).$(9) ).

Sometimes, ${\pi }_0(\mathit{\theta }\mid \tau )$ is not strictly a decreasing function of its component, ${\theta }_i^2$, for $i=1,\dots ,p$, but it could be a decreasing function of the components of some given orthogonal transformation. The following lemma shows that such cases also work.

Lemma 3.6

Consider the model (1(1) $\begin{array}{rl}\mathit{y}\mid \mathit{\theta },\tau & \sim N_p(\mathit{\theta },{\tau }^{-1}{\mathit{I}}_p),\text{independently},\\ w\mid \tau & \sim {\chi }_m^2/\tau ,\end{array}$(1) ). Suppose there is an orthogonal matrix $\mathit{H},$ such that $\mathit{u}=\mathit{H}\mathit{\theta }=(u_1,\dots ,u_p{)}^{\prime },$ and ${\pi }_0(\mathit{\theta }\mid \tau )$ is a decreasing function of $u_i^2,$ for $i=1,\dots ,p,$ then Condition 7 holds.

Proof.

It is easy to verify that the Jacobian of the transformation $\mathit{u}=\mathit{H}\mathit{\theta }=(u_1,\dots ,u_p{)}^{\prime }$ is $J=|\mathrm{\partial }\mathit{\theta }/\mathrm{\partial }\mathit{u}|=1$, and $\parallel \mathit{\theta }\parallel =\parallel \mathit{u}\parallel$. Note that $\begin{array}{rl}\parallel {\mathit{u}}_1\parallel & =∥{\int }_0^{\mathrm{\infty }}{\int }_{\mathrm{\Omega }}{f}_1({\mathit{H}}^{\prime }\mathit{y}\mid \mathit{u},\tau )f_2(w\mid \tau )\\ & \times \phantom{{\int }_0^{\mathrm{\infty }}}{\mathrm{\nabla }}_{\mathit{u}}{\pi }_0({\mathit{H}}^{\prime }\mathit{u}\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\mathit{u}\mathrm{d}\tau ∥,\end{array}$ where $f_1({\mathit{H}}^{\prime }\mathit{y}\mid \mathit{u},\tau )$ is the normal density function of ${\mathit{H}}^{\prime }\mathit{y}$ with mean $\mathit{u}$ and variance ${\tau }^{-1}{\mathit{I}}_p$, and $\mathrm{\Omega }=\{\mathit{u}:\parallel \mathit{u}\parallel \le 1\}$. Similarly, we have $\begin{array}{rl}{z}_1& ={\int }_0^{\mathrm{\infty }}{\int }_{\mathrm{\Omega }}\tau f_1({\mathit{H}}^{\prime }\mathit{y}\mid \mathit{u},\tau )f_2(w\mid \tau )\\ & \times {\pi }_0({\mathit{H}}^{\prime }\mathit{u}\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\mathit{u}\mathrm{d}\tau .\end{array}$ Since ${\pi }_0({\mathit{H}}^{\prime }\mathit{u}\mid \tau )$ s a decreasing function of $u_i^2$, for $i=1,\dots ,p,$ from Lemma 3.4, for any $\mathit{y}$ and w, we have $\begin{array}{rl}& ∥{\int }_0^{\mathrm{\infty }}{\int }_{\mathrm{\Omega }}{f}_1({\mathit{H}}^{\prime }\mathit{y}\mid \mathit{u},\tau )f_2(w\mid \tau )\\ & \times \phantom{{\int }_0^{\mathrm{\infty }}}{\mathrm{\nabla }}_{\mathit{u}}{\pi }_0({\mathit{H}}^{\prime }\mathit{u}\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\mathit{u}\mathrm{d}\tau ∥\\ & \le \parallel {\mathit{H}}^{\prime }\mathit{y}\parallel {\int }_0^{\mathrm{\infty }}{\int }_{\mathrm{\Omega }}\tau f_1({\mathit{H}}^{\prime }\mathit{y}\mid \mathit{u},\tau )f_2(w\mid \tau )\\ & \times {\pi }_0({\mathit{H}}^{\prime }\mathit{u}\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\mathit{u}\mathrm{d}\tau ,\end{array}$ i.e. $\parallel {\mathit{u}}_1\parallel \le z_1\parallel \mathit{y}\parallel$. With the same argument as above, we have $\parallel {\mathit{u}}_2\parallel /z_{2j}\le \parallel \mathit{y}\parallel ,$ for any $j=1,2,\dots$.

Accordingly, we get the following result.

Theorem 3.7

For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ), assume that the prior densities ${\pi }_0(\mathit{\theta }\mid \tau )$ and ${\pi }_1\left(\tau \right)$ satisfy Conditions 1-6. If there is an orthogonal matrix $\mathit{H},$ such that $\mathit{u}=\mathit{H}\mathit{\theta }=(u_1,\dots ,u_p{)}^{\prime },$ and ${\pi }_0(\mathit{\theta }\mid \tau )$ is a decreasing function of $u_i^2,$ for $i=1,\dots ,p,$ the estimator ${\mathit{\delta }}_B\left(\mathit{y}\right)$ in (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) for $\mathit{\theta }$ is admissible under the normalised squared error loss (9(9) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }{\mathit{X}}^{\prime }\mathit{X}(\hat{\mathit{\theta }}-\mathit{\theta }).$(9) ).

In the next two sections, we will apply the above results to a 2-level and a 3-level hierarchical model, with unknown variance and a standard class of hierarchical priors.

4. Admissibility for a 2-level hierarchical model

4.1. g-Prior

For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ), we consider the following class of hierarchical prior for $(\mathit{\theta },\tau )$, (22) $(\mathit{\theta }\mid g,\tau )\sim N_p(\mathbf{0},g{\tau }^{-1}({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}),{\pi }_1(\tau )\propto \frac{1}{{\tau }^k},$(22) where $k\ge 0$. Zellner (1986) proposed this form of the conjugate Normal-Gamma family with k=1. Many authors followed his work, for example, Eaton (1989), Berger, Pericchi, and Varshavsky (1998), Liang, Paulo, Molina, Clyde, and Berger (2008) and Bayarri, Berger, Forte, and Garcła-Donato (2012). From the perspective of model selection, g acts as a dimensionality penalty (Liang et al., 2008). For the choice of g, we study two cases:

Case 1. g is a known positive constant.

Recommendations for g have included the following: Kass & Wasserman's (1995) unit information prior (g=n), Foster & George's (1994) risk inflation criterion ($g=p^2$), Fernández, Ley, & Steel's (2001) benchmark prior ($g=max(n,p^2)$) and so on.

Case 2. g is an unknown parameter, and the prior of g is ${\pi }_2\left(g\right)$.

By integrating out the latent variable g, one can get the conditional prior of $\mathit{\theta }$ given $\tau >0$, (23) $\begin{array}{rl}{\pi }_0(\mathit{\theta }\mid \tau )& ={\int }_0^{\mathrm{\infty }}{\left(\frac{\tau }{2g\pi }\right)}^{p/2}\\ & \times \exp \left\{-\frac{\tau {\mathit{\theta }}^{\prime }{\mathit{X}}^{\prime }\mathit{X}\mathit{\theta }}{2g}\right\}{\pi }_2\left(g\right)\mathrm{d}g,\end{array}$(23) which can be represented as a mixture of g priors.

For Case 2, some priors ${\pi }_2\left(g\right)$ have been previously considered. Here are two examples.

Example 4.1

$\text{Inv-Gamma}(v,c)$, i.e. (24) ${\pi }_2\left(g\right)=\frac{c^v}{\mathrm{\Gamma }\left(v\right)}{g}^{-(v+1)}{\mathrm{e}}^{-c/g}.$(24) As discussed by Berger and Strawderman (1996), it results in the multivariate t-prior for $\mathit{\theta }$ given $\tau >0$, namely (25) ${\pi }_0(\mathit{\theta }\mid \tau )\propto {\tau }^{p/2}{\left[1+\frac{\tau }{2c}{\mathit{\theta }}^{\prime }\left({\mathit{X}}^{\prime }\mathit{X}\right)\mathit{\theta }\right]}^{-(p/2+v)}.$(25) Zellner-Siow (1980) studied the multivariate cauchy prior for $\mathit{\theta }$, which is one special case of (25(25) ${\pi }_0(\mathit{\theta }\mid \tau )\propto {\tau }^{p/2}{\left[1+\frac{\tau }{2c}{\mathit{\theta }}^{\prime }\left({\mathit{X}}^{\prime }\mathit{X}\right)\mathit{\theta }\right]}^{-(p/2+v)}.$(25) ) with $v=1/2$ and $c=n/2$.

Example 4.2

Robust prior (Bayarri et al., 2012): (26) $\begin{array}{rl}{\pi }_2\left(g\right)& =h_1\left[h_3\right(h_2+p\left){]}^{h_1}\right(h_2+g{)}^{-(h_1+1)}\\ & \times 1_{\{g>h_3(h_2+p)-h_2\}}\\ & \propto (h_2+g{)}^{-(h_1+1)}{1}_{\{g>h_3(h_2+p)-h_2\}},\end{array}$(26)

where $h_1>0$, $h_2>0$, and $h_3\ge h_2/(h_2+p)$. The prior (26(26) $\begin{array}{rl}{\pi }_2\left(g\right)& =h_1\left[h_3\right(h_2+p\left){]}^{h_1}\right(h_2+g{)}^{-(h_1+1)}\\ & \times 1_{\{g>h_3(h_2+p)-h_2\}}\\ & \propto (h_2+g{)}^{-(h_1+1)}{1}_{\{g>h_3(h_2+p)-h_2\}},\end{array}$(26) ) has its origins in the robust prior introduced by Strawderman (1971), Berger (1980) and Berger (1985). As Bayarri et al. (2012) discussed, the priors proposed by Liang et al. (2008) are particular cases with $h_1=\frac{1}{2},h_2=1,h_3=1/(1+p)$ (the hyper-g prior) and $h_1=\frac{1}{2},h_2=p,h_3=\frac{1}{2}$ (the hyper-g/n prior). The prior in Cui and George (2008) has $h_1=1,h_2=1,h_3=1/(1+p)$.

For the robust prior (26(26) $\begin{array}{rl}{\pi }_2\left(g\right)& =h_1\left[h_3\right(h_2+p\left){]}^{h_1}\right(h_2+g{)}^{-(h_1+1)}\\ & \times 1_{\{g>h_3(h_2+p)-h_2\}}\\ & \propto (h_2+g{)}^{-(h_1+1)}{1}_{\{g>h_3(h_2+p)-h_2\}},\end{array}$(26) ), it is not straightforward to obtain the closed form of the marginal conditional prior for $\mathit{\theta }$ given τ. Alternatively, we attempt to get the boundary of the marginal density of $\mathit{\theta }$ given τ.

Lemma 4.3

Define (27) $f\left(u\right)\equiv {\int }_0^{\mathrm{\infty }}\frac{1}{v^{r_1}(v+c{)}^{r_2}}\exp \left(-\frac{u}{2v}\right)\mathrm{d}v,$(27) where $r_1>1,$ $r_2\ge 0$ and $c>0,$ then there are two positive constants $C_1$ and $C_2,$ such that $\frac{C_1}{u^{r_1-1}(1+u{)}^{r_2}}\le f\left(u\right)\le \frac{C_2}{u^{r_1-1}(1+u{)}^{r_2}},$ for any u>0.

The proof is given in the Appendix. Applying this lemma to (23(23) $\begin{array}{rl}{\pi }_0(\mathit{\theta }\mid \tau )& ={\int }_0^{\mathrm{\infty }}{\left(\frac{\tau }{2g\pi }\right)}^{p/2}\\ & \times \exp \left\{-\frac{\tau {\mathit{\theta }}^{\prime }{\mathit{X}}^{\prime }\mathit{X}\mathit{\theta }}{2g}\right\}{\pi }_2\left(g\right)\mathrm{d}g,\end{array}$(23) ), the resulting prior for $\mathit{\theta }$ given τ for robust prior (26(26) $\begin{array}{rl}{\pi }_2\left(g\right)& =h_1\left[h_3\right(h_2+p\left){]}^{h_1}\right(h_2+g{)}^{-(h_1+1)}\\ & \times 1_{\{g>h_3(h_2+p)-h_2\}}\\ & \propto (h_2+g{)}^{-(h_1+1)}{1}_{\{g>h_3(h_2+p)-h_2\}},\end{array}$(26) ) with $h_3=h_2/(h_2+p)$ has the boundary (28) $\begin{array}{rl}& \frac{C_1\tau }{\left({\mathit{\theta }}^{\prime }{\mathit{X}}^{\prime }\mathit{X}\mathit{\theta }{)}^{p/2-1}\right(1+\tau {\mathit{\theta }}^{\prime }{\mathit{X}}^{\prime }\mathit{X}\mathit{\theta }{)}^{h_1+1}}\le {\pi }_0(\mathit{\theta }\mid \tau )\\ & \le \frac{C_2\tau }{\left({\mathit{\theta }}^{\prime }{\mathit{X}}^{\prime }\mathit{X}\mathit{\theta }{)}^{p/2-1}\right(1+\tau {\mathit{\theta }}^{\prime }{\mathit{X}}^{\prime }\mathit{X}\mathit{\theta }{)}^{h_1+1}},\end{array}$(28) where p>2.

4.2. Admissibility

We apply the results in Section 3 to determine when the hierarchical priors (22(22) $(\mathit{\theta }\mid g,\tau )\sim N_p(\mathbf{0},g{\tau }^{-1}({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}),{\pi }_1(\tau )\propto \frac{1}{{\tau }^k},$(22) ) result in admissible estimators of $\mathit{\theta }$ under the normalised squared error loss (9(9) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }{\mathit{X}}^{\prime }\mathit{X}(\hat{\mathit{\theta }}-\mathit{\theta }).$(9) ).

Theorem 4.4

$($Case $1)$ For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ) under the hierarchical prior (22(22) $(\mathit{\theta }\mid g,\tau )\sim N_p(\mathbf{0},g{\tau }^{-1}({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}),{\pi }_1(\tau )\propto \frac{1}{{\tau }^k},$(22) ) with a given g. If $0\le k<1,$ the estimator ${\mathit{\delta }}_B\left(\mathit{y}\right)$ in (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) for $\mathit{\theta }$ is admissible under the normalised squared error loss (9(9) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }{\mathit{X}}^{\prime }\mathit{X}(\hat{\mathit{\theta }}-\mathit{\theta }).$(9) ).

The proof of Theorem 4.4 is similar to the proof of Theorem 4.5 later, thus it is omitted. As discussed by George and Foster (2000), the choice of g effectively controls model selection, with large g typically concentrating the prior on parsimonious models with a few large coefficients, whereas small g tends to concentrate the prior on saturated models with small coefficients. Herein, we consider Case 1 from the perspective of admissibility, not the model selection. From Theorem 4.4, the choice of fix g has no effect on the admissibility of estimators ${\mathit{\delta }}_B\left(\mathit{y}\right)$ of $\mathit{\theta }$.

Next, we consider Case 2. The prior density of g satisfies the following conditions:

• Condition A1. ${\pi }_2\left(g\right)$ is a continuous function in $(0,\mathrm{\infty })$;

• Condition A2. $\mathrm{\exists }a\in \mathbb{R}$, ${\pi }_2\left(g\right)=O\left(g^a\right)$, as $g\to 0$;

• Condition A3. $\mathrm{\exists }b\ge 0$, ${\pi }_2\left(g\right)\sim C{g}^{-b}$, as $g\to \mathrm{\infty }$ for some constant C>0.

Clearly, two examples of ${\pi }_2\left(g\right)$ discussed in Section 4.1 satisfy Condition A1–A3 with appropriate a and b.

Theorem 4.5

$($Case $2)$ For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ) with the hierarchical prior (22(22) $(\mathit{\theta }\mid g,\tau )\sim N_p(\mathbf{0},g{\tau }^{-1}({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}),{\pi }_1(\tau )\propto \frac{1}{{\tau }^k},$(22) ), assume ${\pi }_2\left(g\right)$ satisfies Condition A1–A3. If $0\le k<1,$ a>k−1 and $k+b>3,$ the estimator ${\mathit{\delta }}_B\left(\mathit{y}\right)$ in (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) for $\mathit{\theta }$ is admissible under the normalised squared error loss (9(9) $L(\mathit{\theta },\tau ;\hat{\mathit{\theta }})=\tau (\hat{\mathit{\theta }}-\mathit{\theta }{)}^{\prime }{\mathit{X}}^{\prime }\mathit{X}(\hat{\mathit{\theta }}-\mathit{\theta }).$(9) ).

Proof.

It is convenient to write ${\mathit{X}}^{\prime }\mathit{X}={\mathit{H}}^{\prime }\mathit{D}\mathit{H}$, where $\mathit{H}$ is the matrix of eigenvectors corresponding to $\mathit{D}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(d_1,d_2,\dots ,d_p)$ with $d_1\ge \cdots \ge d_p$. Define $\mathit{u}=\mathit{H}\mathit{\theta }=(u_1,\dots ,u_p{)}^{\prime }$. From (23(23) $\begin{array}{rl}{\pi }_0(\mathit{\theta }\mid \tau )& ={\int }_0^{\mathrm{\infty }}{\left(\frac{\tau }{2g\pi }\right)}^{p/2}\\ & \times \exp \left\{-\frac{\tau {\mathit{\theta }}^{\prime }{\mathit{X}}^{\prime }\mathit{X}\mathit{\theta }}{2g}\right\}{\pi }_2\left(g\right)\mathrm{d}g,\end{array}$(23) ), the conditional prior of $\mathit{\theta }$ given $\tau >0$ is $\begin{array}{rl}{\pi }_0(\mathit{\theta }\mid \tau )& ={\int }_0^{\mathrm{\infty }}{\left(\frac{\tau }{2g\pi }\right)}^{p/2}\\ & \times \exp \left\{-\frac{\tau }{2g}\sum _{i=1}^p\frac{u_i^2}{d_i}\right\}{\pi }_2\left(g\right)\mathrm{d}g,\end{array}$ which is a decreasing function of $u_i^2$, for $i=1,\dots ,p$. From Theorem 3.7, we just need to verify Condition 1–6. For Condition 1, $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{1}{\tau }\frac{{\pi }_0\left(\mathit{\theta }|\tau \right)}{\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2)}{\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & ={\int }_{S^c}{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\frac{1}{\tau }\frac{{\pi }_0(\mathit{\theta }\mid g,\tau )}{\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2)}\\ & \times {\pi }_1\left(\tau \right){\pi }_2\left(g\right)\mathrm{d}\tau \mathrm{d}g\mathrm{d}\mathit{\theta }\\ & \le {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{g^{-p/2}}{\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2)}{\pi }_2\left(g\right)\\ & \times \left[{\int }_0^{\mathrm{\infty }}{\tau }^{p/2-k-1}\exp \left(-\frac{d_p\tau \parallel \mathit{\theta }{\parallel }^2}{2g}\right)\mathrm{d}\tau \right]\mathrm{d}g\mathrm{d}\mathit{\theta }.\end{array}$ If $k<p/2$, there is a positive constant C, such that $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{1}{\tau }\frac{{\pi }_0(\mathit{\theta }\mid \tau )}{\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2)}{\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C{\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{g^{-p/2}}{\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2)}\\ & \times {\pi }_2\left(g\right){\left(\frac{\parallel \mathit{\theta }{\parallel }^2}{g}\right)}^{-p/2+k}\mathrm{d}g\mathrm{d}\mathit{\theta }\\ & =C{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}\log (\parallel \mathit{\theta }\parallel \vee 2)}\mathrm{d}\mathit{\theta }\\ & \times {\int }_0^{\mathrm{\infty }}{g}^{-k}{\pi }_2\left(g\right)\mathrm{d}g.\end{array}$ By polar coordinate transformation $r=\parallel \mathit{\theta }\parallel$, the integration over $\mathit{\theta }$ becomes $\begin{array}{rl}& {\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}\log (\parallel \mathit{\theta }\parallel \vee 2)}\mathrm{d}\mathit{\theta }\\ & ={\int }_1^{\mathrm{\infty }}\frac{1}{r^{3-2k}\log (r\vee 2)}\mathrm{d}r,\end{array}$ which is finite if $3-2k\ge 1$, i.e. $k\le 1.$ Since ${\pi }_2\left(g\right)$ satisfies Condition A1-A3, there are some postive constants $N_0<N_1<N_2$, $C_1$ and $C_2$ such that $\begin{array}{rl}{\int }_0^{\mathrm{\infty }}{g}^{-k}{\pi }_2\left(g\right)\mathrm{d}g& ={\int }_0^{N_0}{g}^{-k}{\pi }_2\left(g\right)\mathrm{d}g\\ & +{\int }_{N_0}^{N_1}{g}^{-k}{\pi }_2\left(g\right)\mathrm{d}g\\ & +{\int }_{N_1}^{\mathrm{\infty }}{g}^{-k}{\pi }_2\left(g\right)\mathrm{d}g\\ & \le C_1{\int }_0^{N_0}{g}^{a-k}\mathrm{d}g\\ & +{\int }_{N_0}^{N_1}{g}^{-k}{\pi }_2\left(g\right)\mathrm{d}g\\ & +C_2{\int }_{N_1}^{\mathrm{\infty }}{g}^{-k-b}\mathrm{d}g,\end{array}$ which is finite if a>k−1, and k+b>1.

For Condition 2, $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\le {\int }_{S^c}{\int }_0^{\mathrm{\infty }}{g}^{-p/2}{\pi }_2\left(g\right)\\ & \times \left[{\int }_0^{\mathrm{\infty }}{\tau }^{p/2-k}\exp \left(-\frac{d_p\tau \parallel \mathit{\theta }{\parallel }^2}{2g}\right)\mathrm{d}\tau \right]\mathrm{d}g\mathrm{d}\mathit{\theta }\\ & =\frac{\mathrm{\Gamma }(1+p/2-k)}{(d_p/2{)}^{1+p/2-k}}{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\\ & \times {\int }_0^{\mathrm{\infty }}{g}^{1-k}{\pi }_2\left(g\right)\mathrm{d}g,\end{array}$ which is finite if $0\le k<1$, a>k−2 and k+b>2.

For Condition 3, $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\tau \parallel \mathit{\theta }{\parallel }^2{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\parallel \mathit{\theta }{\parallel }^2{g}^{-p/2}{\pi }_2\left(g\right)\\ & \times \left[{\int }_0^{\mathrm{\infty }}{\tau }^{1+p/2-k}\exp \left(-\frac{d_p\tau \parallel \mathit{\theta }{\parallel }^2}{2g}\right)\mathrm{d}\tau \right]\mathrm{d}g\mathrm{d}\mathit{\theta }\\ & =\frac{\mathrm{\Gamma }(2+p/2-k)}{(d_p/2{)}^{2+p/2-k}}{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\\ & \times {\int }_0^{\mathrm{\infty }}{g}^{2-k}{\pi }_2\left(g\right)\mathrm{d}g,\end{array}$ which is finite if $0\le k<1$, a>k−3 and k+b>3.

For Condition 4, note that $\begin{array}{rl}{\mathrm{\nabla }}_{\mathit{\theta }}{\pi }_0(\mathit{\theta }\mid \tau )& =-{\mathit{X}}^{\prime }\mathit{X}\mathit{\theta }{\tau }^{p/2+1}{\int }_0^{\mathrm{\infty }}{g}^{-(p/2+1)}\\ & \times \exp \left(-\frac{\tau {\mathit{\theta }}^{\prime }{\mathit{X}}^{\prime }\mathit{X}\mathit{\theta }}{2g}\right){\pi }_2\left(g\right)\mathrm{d}g.\end{array}$ By Cauchy-Schwarz inequality, it yields $\begin{array}{rl}\parallel {\mathrm{\nabla }}_{\mathit{\theta }}{\pi }_0(\mathit{\theta }\mid \tau ){\parallel }^2& \le d_1^2\parallel \mathit{\theta }{\parallel }^2{\tau }^{p+2}{\int }_0^{\mathrm{\infty }}{g}^{-p/2}\\ & \times \exp \left(-\frac{\tau {\mathit{\theta }}^{\prime }{\mathit{X}}^{\prime }\mathit{X}\mathit{\theta }}{2g}\right){\pi }_2\left(g\right)\mathrm{d}g\\ & \times {\int }_0^{\mathrm{\infty }}{g}^{-(p/2+2)}\exp \left(-\frac{\tau {\mathit{\theta }}^{\prime }{\mathit{X}}^{\prime }\mathit{X}\mathit{\theta }}{2g}\right)\\ & \times {\pi }_2\left(g\right)\mathrm{d}g\\ & \le d_1^2\parallel \mathit{\theta }{\parallel }^2{\tau }^{p/2+2}{\pi }_0(\mathit{\theta }\mid \tau )\\ & \times {\int }_0^{\mathrm{\infty }}{g}^{-(p/2+2)}\exp \left(-\frac{d_p\tau \parallel \mathit{\theta }{\parallel }^2}{2g}\right)\\ & \times {\pi }_2\left(g\right)\mathrm{d}g.\end{array}$ Therefore, $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{1}{\tau }\frac{\parallel {\mathrm{\nabla }}_{\mathit{\theta }}{\pi }_0(\mathit{\theta }\mid \tau ){\parallel }^2}{{\pi }_0(\mathit{\theta }\mid \tau )}{\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le d_1^2{\int }_{S^c}{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\parallel \mathit{\theta }{\parallel }^2{\tau }^{p/2+1}{g}^{-(p/2+2)}\\ & \times \exp \left(-\frac{d_p\tau \parallel \mathit{\theta }{\parallel }^2}{2g}\right){\pi }_2\left(g\right)\mathrm{d}g{\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & =d_1^2{\int }_{S^c}{\int }_0^{\mathrm{\infty }}\parallel \mathit{\theta }{\parallel }^2{g}^{-(p/2+2)}{\pi }_2\left(g\right)\\ & \times \left[{\int }_0^{\mathrm{\infty }}{\tau }^{p/2+1-k}\exp \left(-\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{2g}\right)\mathrm{d}\tau \right]\mathrm{d}g\mathrm{d}\mathit{\theta }\\ & =\frac{\mathrm{\Gamma }(2+p/2-k)}{(d_p/2{)}^{2+p/2-k}}{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\\ & \times {\int }_0^{\mathrm{\infty }}{g}^{-k}{\pi }_2\left(g\right)\mathrm{d}g,\end{array}$ which is finite if $0\le k<1$, a>k−1, and k+b>1.

For Condition 5, $\begin{array}{rl}& {\int }_{\parallel \mathit{\theta }{\parallel }^2<B}{\int }_{\tau <B}{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le {\int }_0^{\mathrm{\infty }}{\int }_0^B{g}^{-p/2}{\tau }^{p/2-k}\\ & \times \left[{\int }_{\parallel \mathit{\theta }{\parallel }^2<B}\exp \left(-\frac{d_p\tau \parallel \mathit{\theta }{\parallel }^2}{2g}\right)\mathrm{d}\mathit{\theta }\right]{\pi }_2\left(g\right)\mathrm{d}\tau \mathrm{d}g.\end{array}$ By polar coordinate transformation $r=\parallel \mathit{\theta }{\parallel }^2$, the integration over $\mathit{\theta }$ becomes $\begin{array}{rl}& {\int }_{\parallel \mathit{\theta }{\parallel }^2<B}\exp \left(-\frac{d_p\tau \parallel \mathit{\theta }{\parallel }^2}{2g}\right)\mathrm{d}\mathit{\theta }\\ & ={\int }_0^B{r}^{p/2-1}\exp \left(-\frac{d_p\tau r}{2g}\right)\mathrm{d}r\\ & \le C{\tau }^{-p/2}{g}^{p/2}.\end{array}$ Therefore, $\begin{array}{rl}& {\int }_{\parallel \mathit{\theta }{\parallel }^2<B}{\int }_{\tau <B}{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C{\int }_0^B{\tau }^{-k}\mathrm{d}\tau {\int }_0^{\mathrm{\infty }}{\pi }_2\left(g\right)\mathrm{d}g,\end{array}$ which is finite if $0\le k<1$, a>−1, and b>1.

Combining these restrictions, we can find that when $0\le k<1$, a>k−1 and k+b>3, Conditions 1–5 hold. As Han (2009) discussed, Condition 6 is very mild. Proceeding in an analogous way on page 47 of Han (2009), Condition 6 holds. By Theorem 3.7, the estimator ${\mathit{\delta }}_B\left(\mathit{y}\right)$ in (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) for $\mathit{\theta }$ is admissible.

We are also interested in admissible estimators under Inv-Gamma and robust prior for g. Using Theorem 4.5, we have the following results.

Theorem 4.6

For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ) with the hierarchical prior (22(22) $(\mathit{\theta }\mid g,\tau )\sim N_p(\mathbf{0},g{\tau }^{-1}({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}),{\pi }_1(\tau )\propto \frac{1}{{\tau }^k},$(22) ), assume ${\pi }_2\left(g\right)$ is $\text{Inv-Gamma}(v,c)$. If $0\le k<1,$ and $v>2-k,$ the estimator ${\mathit{\delta }}_B\left(\mathit{y}\right)$ in (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) for $\mathit{\theta }$ is admissible.

Proof.

By Theorem 4.5 with any constant a>k−1 and b=v+1, the result holds.

Theorem 4.7

For the model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ) with the hierarchical prior (22(22) $(\mathit{\theta }\mid g,\tau )\sim N_p(\mathbf{0},g{\tau }^{-1}({\mathit{X}}^{\prime }\mathit{X}{)}^{-1}),{\pi }_1(\tau )\propto \frac{1}{{\tau }^k},$(22) ), assume ${\pi }_2\left(g\right)$ is robust prior (26(26) $\begin{array}{rl}{\pi }_2\left(g\right)& =h_1\left[h_3\right(h_2+p\left){]}^{h_1}\right(h_2+g{)}^{-(h_1+1)}\\ & \times 1_{\{g>h_3(h_2+p)-h_2\}}\\ & \propto (h_2+g{)}^{-(h_1+1)}{1}_{\{g>h_3(h_2+p)-h_2\}},\end{array}$(26) ). If $0\le k<1,$ and $h_1>2-k,$ the estimator ${\mathit{\delta }}_B\left(\mathit{y}\right)$ in (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) for $\mathit{\theta }$ is admissible.

Proof.

From Theorem 4.5 with a=0 and $b=h_1+1$, the proof is completed.

5. Admissibility for a 3-level hierarchical model

We also study a 3-level hierarchical model and determine which elements of the hierarchical prior class lead to admissible estimators of the $\mathit{\theta }$ under normalised squared error loss.

5.1. The model and priors

Consider the following 3-level hierarchical model (29) $\begin{array}{rl}& Level1:(\mathit{y}\mid \mathit{\theta },\tau )\sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n);\\ & Level2:(\mathit{\theta }\mid \mathit{\beta },\tau ,g)\sim N_p(\mathit{Z}\mathit{\beta },g{\tau }^{-1}\mathit{A});\\ & Level3:(\mathit{\beta }\mid \lambda ,\tau )\sim N_s(\mathbf{0}\mathbf{,}\lambda {\tau }^{\mathbf{-}\mathbf{1}}\mathit{B}\mathbf{)}\mathbf{,}\end{array}$(29) where $\mathit{Z}$ is a given $p\times s$ matrix with full rank s, $\mathit{\beta }$ is the $s\times 1$ unknown vector, $\mathit{A}$ and $\mathit{B}$ are a $p\times p$ and $s\times s$ known covariate matrix, respectively, and λ is an unknown hyperparameter. To simplify the computation, without loss of generality, we set $\mathit{A}={\mathit{I}}_p$ and $\mathit{B}={\mathit{I}}_s$.

Assume ${\pi }_1\left(\tau \right)\propto {\tau }^{-k}$, and the prior of g satisfies the Condition A1–A3. The prior ${\pi }_3\left(\lambda \right)$ satisfies the following conditions.

• Condition B1. ${\pi }_3\left(\lambda \right)$ is a continuous function in $(0,\mathrm{\infty })$;

• Condition B2. $\mathrm{\exists }{c}_1$, ${\pi }_3\left(\lambda \right)=O\left({\lambda }^{c_1}\right)$, as $\lambda \to 0$;

• Condition B3. $\mathrm{\exists }{c}_2\ge 0$, ${\pi }_3\left(\lambda \right)\sim C{\lambda }^{-c_2}$ as $\lambda \to \mathrm{\infty }$ for some constant C>0.

5.2. Admissibility

The following lemma is needed.

Lemma 5.1

For the 3-level hierarchical model (29(29) $\begin{array}{rl}& Level1:(\mathit{y}\mid \mathit{\theta },\tau )\sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n);\\ & Level2:(\mathit{\theta }\mid \mathit{\beta },\tau ,g)\sim N_p(\mathit{Z}\mathit{\beta },g{\tau }^{-1}\mathit{A});\\ & Level3:(\mathit{\beta }\mid \lambda ,\tau )\sim N_s(\mathbf{0}\mathbf{,}\lambda {\tau }^{\mathbf{-}\mathbf{1}}\mathit{B}\mathbf{)}\mathbf{,}\end{array}$(29) ), assume ${\pi }_1\left(\tau \right)\propto {\tau }^{-k},$ ${\pi }_2\left(g\right)$ satisfies Condition A1–A3, and ${\pi }_3\left(\lambda \right)$ satisfies Condition B1–B3. Then (30) $\begin{array}{rl}& {\pi }_0(\mathit{\theta }\mid \tau )\propto {\tau }^{p/2}\iint \exp \left[-\frac{1}{2}\tau {\mathit{\theta }}^{\prime }(g{\mathit{I}}_p+\lambda \mathit{Z}{\mathit{Z}}^{\prime }{)}^{-1}\mathit{\theta }\right]\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)/2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda .\end{array}$(30)

Proof.

Note that $\begin{array}{rl}{\pi }_0(\mathit{\theta }\mid \tau )& \propto {\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\int }_{{\mathbb{R}}^s}{\pi }_0(\mathit{\theta }\mid \mathit{\beta },\tau ,g)\\ & \pi (\mathit{\beta }\mid \lambda ){\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}\mathit{\beta }\mathrm{d}g\mathrm{d}\lambda \\ & \propto {\tau }^{(p+s)/2}{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}{\int }_{{\mathbb{R}}^s}\\ & \times \exp \left(-\frac{\tau \parallel \mathit{\theta }-\mathit{Z}\mathit{\beta }{\parallel }^2}{2g}-\frac{\tau \parallel \mathit{\beta }{\parallel }^2}{2\lambda }\right)\\ & \times g^{-p/2}{\lambda }^{-s/2}{\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}\mathit{\beta }\mathrm{d}g\mathrm{d}\lambda .\end{array}$ Define ${\mathit{\beta }}_0=g^{-1}({\lambda }^{-1}{\mathit{I}}_s+g^{-1}{\mathit{Z}}^{\prime }\mathit{Z}{)}^{-1}{\mathit{Z}}^{\prime }\mathit{\theta },$ we have $\begin{array}{rl}\frac{\tau \parallel \mathit{\theta }-\mathit{Z}\mathit{\beta }{\parallel }^2}{g}+\frac{\tau \parallel \mathit{\beta }{\parallel }^2}{\lambda }& =\tau (\mathit{\beta }-{\mathit{\beta }}_0{)}^{\prime }\\ & \times ({\lambda }^{-1}{\mathit{I}}_s+g^{-1}{\mathit{Z}}^{\prime }\mathit{Z})\\ & \times (\mathit{\beta }-{\mathit{\beta }}_0)+\tau {\mathit{\theta }}^{\prime }\\ & \times (g{\mathit{I}}_p+\lambda \mathit{Z}{\mathit{Z}}^{\prime }{)}^{-1}\mathit{\theta }.\end{array}$ Then the marginal distribution of θ given τ, $\begin{array}{rl}{\pi }_0(\mathit{\theta }\mid \tau )& \propto {\tau }^{p/2}{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}|{\lambda }^{-1}{\mathit{I}}_s+g^{-1}{\mathit{Z}}^{\prime }\mathit{Z}{|}^{-1/2}\\ & \times \exp \left[-\frac{1}{2}\tau {\mathit{\theta }}^{\prime }(g{\mathit{I}}_p+\lambda \mathit{Z}{\mathit{Z}}^{\prime }{)}^{-1}\mathit{\theta }\right]\\ & \times g^{-p/2}{\lambda }^{-s/2}{\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda ,\end{array}$ which is proportional to (30(30) $\begin{array}{rl}& {\pi }_0(\mathit{\theta }\mid \tau )\propto {\tau }^{p/2}\iint \exp \left[-\frac{1}{2}\tau {\mathit{\theta }}^{\prime }(g{\mathit{I}}_p+\lambda \mathit{Z}{\mathit{Z}}^{\prime }{)}^{-1}\mathit{\theta }\right]\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)/2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda .\end{array}$(30) ). The proof is completed.

Theorem 5.2

For the 3-level hierarchical model (29(29) $\begin{array}{rl}& Level1:(\mathit{y}\mid \mathit{\theta },\tau )\sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n);\\ & Level2:(\mathit{\theta }\mid \mathit{\beta },\tau ,g)\sim N_p(\mathit{Z}\mathit{\beta },g{\tau }^{-1}\mathit{A});\\ & Level3:(\mathit{\beta }\mid \lambda ,\tau )\sim N_s(\mathbf{0}\mathbf{,}\lambda {\tau }^{\mathbf{-}\mathbf{1}}\mathit{B}\mathbf{)}\mathbf{,}\end{array}$(29) ), assume ${\pi }_1\left(\tau \right)\propto {\tau }^{-k},$ ${\pi }_2\left(g\right)$ satisfies Condition A1–A3, and ${\pi }_3\left(\lambda \right)$ satisfies Condition B1–B3. Then, the estimator ${\mathit{\delta }}_B\left(\mathit{y}\right)$ in (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) for $\mathit{\theta }$ is admissible if $0\le k<1,$ $b>3-k,$ $c_2>3+(p-s)/2-k,$ and one of the following conditions holds,

1. $p>s,$ $a>(p-s)/2+1,$ $c_1>-1;$

2. $p=s,$ $a>-1,$ $c_1>-1+k;$

3. $p=s,$ $a>-1+k,$ $c_1>-1$.

Proof.

It is convenient to write $\mathit{Z}{\mathit{Z}}^{\prime }={\mathbf{\Gamma }}^{\prime }\mathit{D}\mathbf{\Gamma }$, where $\mathbf{\Gamma }$ is the matrix of eigenvectors corresponding to $\mathit{D}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(z_1,z_2,\dots ,z_p)$ with $z_1\ge \cdots \ge z_p$. Herein, we denote $\mathit{u}=\mathbf{\Gamma }\mathit{\theta }=(u_1,\dots ,u_p{)}^{\prime }$. Therefore, from Lemma 5.1, $\begin{array}{rl}& {\pi }_0(\mathit{\theta }\mid \tau )\propto {\tau }^{p/2}\iint \exp \left[-\frac{\tau }{2}\sum _{i=1}^p(g+z_i\lambda {)}^{-1}{u}_i^2\right]\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda ,\end{array}$ which is a decreasing function of $u_i^2$, for $i=1,\dots ,p$. In addition, there are two positive constant $C_1$ and $C_2$, such that $\begin{array}{rl}& {\pi }_0(\mathit{\theta }\mid \tau )\le C_1{\tau }^{p/2}\iint \exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)/2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda .\end{array}$ For the technical reasons, we first consider Condition 2. Note that $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\le C_1{\int }_{S^c}{\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\\ & \times \left\{{\int }_0^{\mathrm{\infty }}{\tau }^{p/2-k}\exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\mathrm{d}\tau \right\}\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)/2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}\lambda \mathrm{d}g\mathrm{d}\mathit{\theta }\\ & =C_1\frac{\mathrm{\Gamma }(1+p/2-k)}{C_2^{1+p/2-k}}{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\\ & \times \iint (\lambda +g{)}^{1+(p-s)/2-k}{g}^{-(p-s)/2}\\ & \times {\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda .\end{array}$ The integration over $\mathit{\theta }$ is finite if k<1. For simplicity, denote $l=1+(p-s)/2-k$ and $h=(p-s)/2$. If $0\le k<1$, we have l>0.

Note that (31) $\begin{array}{rl}& {\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}(\lambda +g{)}^{1+(p-s)/2-k}{g}^{-(p-s)/2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda \\ & =\left\{{\int }_0^1{\int }_0^1+{\int }_0^1{\int }_0^{\mathrm{\infty }}+{\int }_1^{\mathrm{\infty }}{\int }_0^1+{\int }_1^{\mathrm{\infty }}{\int }_1^{\mathrm{\infty }}\right\}\\ & \times (\lambda +g{)}^l{g}^{-h}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda \\ & \equiv I_1+I_2+I_3+I_4.\end{array}$(31) Clearly, $I_1\le 2^l{\int }_0^1{g}^{-h}{\pi }_2\left(g\right)\mathrm{d}g\times {\int }_0^1{\pi }_3\left(\lambda \right)\mathrm{d}\lambda ,$ which is finite if a>h−1 and $c_1>-1$. Clearly, $I_4\le 2^l{\int }_1^{\mathrm{\infty }}{g}^{l-h}{\pi }_2\left(g\right)\mathrm{d}g\times {\int }_1^{\mathrm{\infty }}{\lambda }^l{\pi }_3\left(\lambda \right)\mathrm{d}\lambda ,$ which is finite if b>1−h+l and $c_2>1+l$. Similarly, it is easy to verify that $I_2+I_3$ is finite if a>h−1, b>1−h+l, $c_1>-1$ and $c_2>1+l$. Therefore, (31(31) $\begin{array}{rl}& {\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}(\lambda +g{)}^{1+(p-s)/2-k}{g}^{-(p-s)/2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda \\ & =\left\{{\int }_0^1{\int }_0^1+{\int }_0^1{\int }_0^{\mathrm{\infty }}+{\int }_1^{\mathrm{\infty }}{\int }_0^1+{\int }_1^{\mathrm{\infty }}{\int }_1^{\mathrm{\infty }}\right\}\\ & \times (\lambda +g{)}^l{g}^{-h}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda \\ & \equiv I_1+I_2+I_3+I_4.\end{array}$(31) ) is finite if $a>(p-s)/2-1,b>2-k,c_1>-1$ and $c_2>2+(p-s)/2-k$.

Similarly, for Condition 3, (32) $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\tau \parallel \mathit{\theta }{\parallel }^2{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C_1^{\prime }{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\\ & \times \iint (\lambda +g{)}^{2+(p-s)/2-k}{g}^{-(p-s)/2}\\ & \times {\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda ,\end{array}$(32) where $C_1^{\prime }$ is a postive constant.

Clearly, $2+(p-s)/2-k>0$ if $0\le k<1$. As in the proof of Condition 2, (32(32) $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\tau \parallel \mathit{\theta }{\parallel }^2{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C_1^{\prime }{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\\ & \times \iint (\lambda +g{)}^{2+(p-s)/2-k}{g}^{-(p-s)/2}\\ & \times {\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda ,\end{array}$(32) ) is finite if $0\le k<1$, $a>(p-s)/2-1,b>3-k,c_1>-1$ and $c_2>3+(p-s)/2-k$.

For Condition 4, from Lemma 5.1, note that $\begin{array}{rl}{\mathrm{\nabla }}_{\mathit{\theta }}{\pi }_0(\mathit{\theta }\mid \tau )& =-{\tau }^{p/2+1}\iint (g{\mathit{I}}_p+\lambda \mathit{Z}{\mathit{Z}}^{\prime }{)}^{-1}\mathit{\theta }\\ & \times \exp \left[-\frac{1}{2}\tau {\mathit{\theta }}^{\prime }(g{\mathit{I}}_p+\lambda \mathit{Z}{\mathit{Z}}^{\prime }{)}^{-1}\mathit{\theta }\right]\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)/2}{\pi }_3(\lambda )\\ & \times {\pi }_2\left(g\right)\mathrm{d}g\mathrm{d}\lambda .\end{array}$ We will consider two cases, i.e. p>s and $p=s,$ respectively. If p>s, $\parallel (g{\mathit{I}}_p+\lambda \mathit{Z}{\mathit{Z}}^{\prime }{)}^{-1}\mathit{\theta }\parallel \le g^{-1}\parallel \mathit{\theta }\parallel$. It yields $\begin{array}{rl}& \parallel {\mathrm{\nabla }}_{\mathit{\theta }}{\pi }_0(\mathit{\theta }\mid \tau ){\parallel }^2\le \parallel \mathit{\theta }{\parallel }^2{\tau }^{p+2}\\ & \times \left(\iint \exp \left[-\frac{1}{2}\tau {\mathit{\theta }}^{\prime }(g{\mathit{I}}_p+\lambda \mathit{Z}{\mathit{Z}}^{\prime }{)}^{-1}\mathit{\theta }\right]\right.\\ & \times {\left.(g+\lambda {)}^{-s/2}{g}^{-(p-s)/2-1}{\pi }_3(\lambda \left){\pi }_2\right(g)\mathrm{d}g\mathrm{d}\lambda \right)}^2\\ & \le \parallel \mathit{\theta }{\parallel }^2{\tau }^{p/2+2}{\pi }_0(\mathit{\theta }\mid \tau )\\ & \times \iint \exp \left[-\frac{1}{2}\tau {\mathit{\theta }}^{\prime }(g{\mathit{I}}_p+\lambda \mathit{Z}{\mathit{Z}}^{\prime }{)}^{-1}\mathit{\theta }\right]\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)/2-2}\\ & \times {\pi }_3\left(\lambda \right){\pi }_2\left(g\right)\mathrm{d}g\mathrm{d}\lambda \\ & \le C_1|\mathit{\theta }{\parallel }^2{\tau }^{p/2+2}{\pi }_0(\mathit{\theta }\mid \tau )\iint \exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)/2-2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda .\end{array}$ In the second step, we apply the Cauchy-Schwartz inequality. Therefore, (33) $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{1}{\tau }\frac{\parallel {\mathrm{\nabla }}_{\mathit{\theta }}{\pi }_0(\mathit{\theta }\mid \tau ){\parallel }^2}{{\pi }_0(\mathit{\theta }\mid \tau )}{\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C_1{\int }_{S^c}\iint \parallel \mathit{\theta }{\parallel }^2\left\{{\int }_0^{\mathrm{\infty }}{\tau }^{1+p/2-k}\right.\\ & \times \left.\exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\mathrm{d}\tau \right\}(g+\lambda {)}^{-s/2}{g}^{-(p-s)/2-2}\\ & \times {\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda \mathrm{d}\mathit{\theta }\\ & =C_1^{\prime }{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\times \iint (\lambda +g{)}^{(p-s)/2+2-k}\\ & \times g^{-(p-s)/2-2}{\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda .\end{array}$(33) As in the proof of Condition 3, (33(33) $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{1}{\tau }\frac{\parallel {\mathrm{\nabla }}_{\mathit{\theta }}{\pi }_0(\mathit{\theta }\mid \tau ){\parallel }^2}{{\pi }_0(\mathit{\theta }\mid \tau )}{\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C_1{\int }_{S^c}\iint \parallel \mathit{\theta }{\parallel }^2\left\{{\int }_0^{\mathrm{\infty }}{\tau }^{1+p/2-k}\right.\\ & \times \left.\exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\mathrm{d}\tau \right\}(g+\lambda {)}^{-s/2}{g}^{-(p-s)/2-2}\\ & \times {\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda \mathrm{d}\mathit{\theta }\\ & =C_1^{\prime }{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\times \iint (\lambda +g{)}^{(p-s)/2+2-k}\\ & \times g^{-(p-s)/2-2}{\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda .\end{array}$(33) ) is finite if $0\le k<1,$ $a>(p-s)/2+1$, b>1−k, $c_1>-1$ and $c_2>3+(p-s)/2-k$.

If p=s, there is a positive constant $C_3$, such that $\parallel (g{\mathit{I}}_p+\lambda \mathit{Z}{\mathit{Z}}^{\prime }{)}^{-1}\mathit{\theta }\parallel \le C_3(g+\lambda {)}^{-1}\parallel \mathit{\theta }\parallel .$ Therefore, using the Cauchy-Schwartz inequality, $\begin{array}{rl}\parallel {\mathrm{\nabla }}_{\mathit{\theta }}{\pi }_0(\mathit{\theta }\mid \tau ){\parallel }^2& \le C_4\parallel \mathit{\theta }{\parallel }^2{\tau }^{p/2+2}{\pi }_0(\mathit{\theta }\mid \tau )\\ & \times \iint \exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\\ & \times (g+\lambda {)}^{-p/2-2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda ,\end{array}$ where $C_4=C_1{C}_3^2$. Thus, (34) $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{1}{\tau }\frac{\parallel {\mathrm{\nabla }}_{\mathit{\theta }}{\pi }_0(\mathit{\theta }\mid \tau ){\parallel }^2}{{\pi }_0(\mathit{\theta }\mid \tau )}{\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C_4{\int }_{S^c}\iint \parallel \mathit{\theta }{\parallel }^2\\ & \times \left\{{\int }_0^{\mathrm{\infty }}{\tau }^{1+p/2-k}\exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\mathrm{d}\tau \right\}\\ & \times (g+\lambda {)}^{-p/2-2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda \mathrm{d}\mathit{\theta }\\ & =C_4^{\prime }{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\\ & \times \iint (\lambda +g{)}^{-k}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda ,\end{array}$(34) where $C_4^{\prime }$ is a positive constant. Note that the integration over $\mathit{\theta }$ is finite if $0\le k<1$.

If $k\ge 0,$ $\begin{array}{rl}& {\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}(\lambda +g{)}^{-k}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda \\ & \le {\int }_0^{\mathrm{\infty }}{\pi }_2\left(g\right)\mathrm{d}g{\int }_0^{\mathrm{\infty }}{\lambda }^{-k}{\pi }_3\left(\lambda \right)\mathrm{d}\lambda ,\end{array}$ which is finite if $a>-1,b>1,c_1>-1+k$ and $c_2>1-k.$ Meanwhile, $\begin{array}{rl}& {\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}(\lambda +g{)}^{-k}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda \\ & \le {\int }_0^{\mathrm{\infty }}{g}^{-k}{\pi }_2\left(g\right)\mathrm{d}g{\int }_0^{\mathrm{\infty }}{\pi }_3\left(\lambda \right)\mathrm{d}\lambda ,\end{array}$ which is finite if $a>-1+k,b>1-k,c_1>-1$ and $c_2>1$.

For Condition 5, note that $\begin{array}{rl}& {\int }_{\parallel \mathit{\theta }{\parallel }^2<B}{\int }_{\tau <B}{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C_1{\int }_0^B\iint {\tau }^{p/2-k}\\ & \times \left\{{\int }_{\parallel \mathit{\theta }{\parallel }^2<B}\exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\mathrm{d}\mathit{\theta }\right\}\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)/2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}\lambda \mathrm{d}g\mathrm{d}\tau .\end{array}$ By polar coordinate transformation $r=\parallel \mathit{\theta }{\parallel }^2$, the integration over $\mathit{\theta }$ becomes $\begin{array}{rl}& {\int }_{\parallel \mathit{\theta }{\parallel }^2<B}\exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\mathrm{d}\mathit{\theta }\\ & ={\int }_0^B{r}^{p/2-1}\exp \left(-C_2\frac{\tau r}{g+\lambda }\right)\mathrm{d}\\ & \le C_5{\tau }^{-p/2}(g+\lambda {)}^{p/2},\end{array}$ where $C_5$ is a positive constant. Therefore, (35) $\begin{array}{rl}& {\int }_{\parallel \mathit{\theta }{\parallel }^2<B}{\int }_{\tau <B}{\pi }_0(\mathit{\theta }\mid \tau ){\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C_5{\int }_0^B{\tau }^{-k}\mathrm{d}\tau \times {\int }_0^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}(\lambda +g{)}^{(p-s)/2}\\ & \times g^{-(p-s)/2}{\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda ,\end{array}$(35) which is finite if $0\le k<1$, $a>(p-s)/2-1,b>1,c_1>-1$ and $c_2>1+(p-s)/2$.

Combining the above results, Conditions 2–5 hold if $(k,a,b,c_1,c_2)$ satisfy the conditions as this theorem states. For Condition 1, (36) $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{1}{\tau }\frac{{\pi }_0(\mathit{\theta }\mid \tau )}{\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2)}{\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C_1{\int }_{S^c}\iint \frac{1}{\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2)}\\ & \times \left\{{\int }_0^{\mathrm{\infty }}{\tau }^{p/2-k-1}\exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\mathrm{d}\tau \right\}\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)2}\\ & \times {\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda \mathrm{d}\mathit{\theta }\\ & \le C_6{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\\ & \times \iint (\lambda +g{)}^{(p-s)/2-k}{g}^{-(p-s)/2}\\ & \times {\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda ,\end{array}$(36) where $C_6$ is a positive constant. If $p=s,$ (36(36) $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{1}{\tau }\frac{{\pi }_0(\mathit{\theta }\mid \tau )}{\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2)}{\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C_1{\int }_{S^c}\iint \frac{1}{\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2)}\\ & \times \left\{{\int }_0^{\mathrm{\infty }}{\tau }^{p/2-k-1}\exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\mathrm{d}\tau \right\}\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)2}\\ & \times {\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda \mathrm{d}\mathit{\theta }\\ & \le C_6{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\\ & \times \iint (\lambda +g{)}^{(p-s)/2-k}{g}^{-(p-s)/2}\\ & \times {\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda ,\end{array}$(36) ) can be proceeded as (34(34) $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{1}{\tau }\frac{\parallel {\mathrm{\nabla }}_{\mathit{\theta }}{\pi }_0(\mathit{\theta }\mid \tau ){\parallel }^2}{{\pi }_0(\mathit{\theta }\mid \tau )}{\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C_4{\int }_{S^c}\iint \parallel \mathit{\theta }{\parallel }^2\\ & \times \left\{{\int }_0^{\mathrm{\infty }}{\tau }^{1+p/2-k}\exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\mathrm{d}\tau \right\}\\ & \times (g+\lambda {)}^{-p/2-2}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda \mathrm{d}\mathit{\theta }\\ & =C_4^{\prime }{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\\ & \times \iint (\lambda +g{)}^{-k}{\pi }_2(g\left){\pi }_3\right(\lambda )\mathrm{d}g\mathrm{d}\lambda ,\end{array}$(34) ). If $p>s,$ it is easy to verify that (36(36) $\begin{array}{rl}& {\int }_{S^c}{\int }_0^{\mathrm{\infty }}\frac{1}{\tau }\frac{{\pi }_0(\mathit{\theta }\mid \tau )}{\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2)}{\pi }_1\left(\tau \right)\mathrm{d}\tau \mathrm{d}\mathit{\theta }\\ & \le C_1{\int }_{S^c}\iint \frac{1}{\parallel \mathit{\theta }{\parallel }^2\log (\parallel \mathit{\theta }\parallel \vee 2)}\\ & \times \left\{{\int }_0^{\mathrm{\infty }}{\tau }^{p/2-k-1}\exp \left(-C_2\frac{\tau \parallel \mathit{\theta }{\parallel }^2}{g+\lambda }\right)\mathrm{d}\tau \right\}\\ & \times (g+\lambda {)}^{-s/2}{g}^{-(p-s)2}\\ & \times {\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda \mathrm{d}\mathit{\theta }\\ & \le C_6{\int }_{S^c}\frac{1}{\parallel \mathit{\theta }{\parallel }^{2+p-2k}}\mathrm{d}\mathit{\theta }\\ & \times \iint (\lambda +g{)}^{(p-s)/2-k}{g}^{-(p-s)/2}\\ & \times {\pi }_2\left(g\right){\pi }_3\left(\lambda \right)\mathrm{d}g\mathrm{d}\lambda ,\end{array}$(36) ) is finite if $0\le k<1,$ $a>(p-s)/2+1,$ $b>3-k,$ $c_2>3+(p-s)/2-k$, $c_1>-1.$ Proceeding in an analogous way on page 47 of Han (2009), Condition 6 also holds. By Theorem 3.7, estimator (11(11) ${\mathit{\delta }}_B\left(\mathit{y}\right)=\left({\mathit{T}}^{\prime }{)}^{-1}{\hat{\mathit{\theta }}}_B\right({\mathit{T}}^{\prime }\tilde{\mathit{y}},w),$(11) ) for $\mathit{\theta }$ are admissible.

6. Comments

In Section 2, we listed the sufficient conditions for admissibility of the estimators of $\mathit{\theta }$ with unknown τ, which was developed by Han (2009). In Section 3, we generalise the sufficient conditions for admissibility and apply these results to the normal linear regression model (2(2) $\mathit{y}\mid \mathit{\theta },\tau \sim N_n(\mathit{X}\mathit{\theta },{\tau }^{-1}{\mathit{I}}_n),$(2) ). We have to admit that those sufficient conditions are still not optimal enough. Sometimes, we can't obtain satisfactory results utilising the conditions directly. In our paper, we consider ${\pi }_1\left(\tau \right)\propto {\tau }^{-k}$ for the prior of τ. The condition of k for admissibility is $0\le k<1.$ Unfortunately, we can't prove the admissibility for the boundary point $k=1,$ which is of great interest since it is the natural extension of Stein's harmonic prior (Stein, 1981) to the unknown variance problem. In follow-up work, we will try to explore the more powerful sufficient conditions for admissibility of the estimators of $\mathit{\theta }$ with unknown τ. One promising method for this problem may be by Blyth's method (Blyth, 1951), discovering an appropriate sequence of finite measures.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The project was supported by the 111 Project of China (No. B14019), the National Natural Science Foundation of China [Grant No. 11671146].

Notes on contributors

Chengyuan Song

Chengyuan Song is a PhD candidate in the College of Statistics, East China Normal University, Shanghai, China. His research interests include Bayesian statistics.

Dongchu Sun

Dr. Dongchu Sun received his PhD. in 1991 from Department of Statistics, Purdue University, under the guidence of Professor James O. Berger.

Appendix

A.1. Proof of Lemma 4.3

To simplify the computation, without loss of generality, we set c=1. Let $x=u/v$, then $\begin{array}{rl}f\left(u\right)& ={\int }_0^{\mathrm{\infty }}\frac{1}{\left(u/x{)}^a\right(u/x+1{)}^b}\frac{u}{x^2}\exp (-x/2)\mathrm{d}x\\ & =u^{1-a}{\int }_0^{\mathrm{\infty }}\frac{x^{a+b-2}}{(u+x{)}^b}\exp (-x/2)\mathrm{d}x.\end{array}$One just needs to consider (A1) $\frac{C_1}{(1+u{)}^b}\le {\int }_0^{\mathrm{\infty }}\frac{x^{a+b-2}}{(u+x{)}^b}\exp (-x/2)\mathrm{d}x\le \frac{C_2}{(1+u{)}^b}.$(A1)

The integral can be written by $\left\{{\int }_0^1+{\int }_1^{\mathrm{\infty }}\right\}\frac{x^{a+b-2}}{(u+x{)}^b}\exp (-x/2)\mathrm{d}x\equiv I_1+I_2.$ For the lower bound of $I_1$ and $I_2$, we have $\begin{array}{rl}{I}_1& \ge {\int }_0^1\frac{x^{a+b-2}}{(u+1{)}^b}\exp (-x/2)\mathrm{d}x=C_1^{\ast }\frac{1}{(1+u{)}^b},\\ I_2& \ge {\int }_1^{\mathrm{\infty }}\frac{x^{a-2}}{(u+1{)}^b}\exp (-x/2)\mathrm{d}x=C_2^{\ast }\frac{1}{(1+u{)}^b},\end{array}$ where $C_1^{\ast }={\int }_0^1{x}^{a+b-2}\exp (-x/2)\mathrm{d}x$ and $C_2^{\ast }={\int }_1^{\mathrm{\infty }}{x}^{a-2}\exp (-x/2)\mathrm{d}x$. For the upper bound of $I_1$ and $I_2$, we have $\begin{array}{rl}{I}_1& \le {\int }_0^1\frac{x^{a+b-2}}{(u+1{)}^b{x}^b}\exp (-x/2)\mathrm{d}x=C_3^{\ast }\frac{1}{(1+u{)}^b},\\ I_2& \le {\int }_1^{\mathrm{\infty }}\frac{x^{a+b-2}}{(u+1{)}^b}\exp (-x/2)\mathrm{d}x=C_4^{\ast }\frac{1}{(1+u{)}^b},\end{array}$ where $C_3^{\ast }={\int }_0^1{x}^{a-2}\exp (-x/2)\mathrm{d}x$ and $C_4^{\ast }={\int }_1^{\mathrm{\infty }}{x}^{a+b-2}\exp (-x/2)\mathrm{d}x$. Therefore, let $C_1=C_1^{\ast }+C_2^{\ast }$ and $C_2=C_3^{\ast }+C_4^{\ast }$. We get (A1(A1) $\frac{C_1}{(1+u{)}^b}\le {\int }_0^{\mathrm{\infty }}\frac{x^{a+b-2}}{(u+x{)}^b}\exp (-x/2)\mathrm{d}x\le \frac{C_2}{(1+u{)}^b}.$(A1) ). The lemma is proved.