Bayes minimax estimation under power priors of location parameters for a wide class of spherically symmetric distributions

Abstract

We complement the results of Fourdrinier, Mezoued and Strawderman in [5] who considered Bayesian estimation of the location parameter $\theta$ of a random vector $X$ having a unimodal spherically symmetric density $f(\|x-\theta\|^{2})$ for a spherically symmetric prior density $\pi(\|\theta\|^{2})$. In [5], expressing the Bayes estimator as $\delta_{\pi}(X)=X+\nabla M(\|X\|^{2})/m(\|X\|^{2})$, where $m$ is the marginal associated to $f(\|x-\theta\|^{2})$ and $M$ is the marginal with respect to $F(\|x-\theta\|^{2})=1/2\int_{\|x-\theta\|^{2}}^{\infty}f(t)\,dt$, it was shown that, under quadratic loss, if the sampling density $f(\|x-\theta\|^{2})$ belongs to the Berger class (i.e. there exists a positive constant $c$ such that $F(t)/f(t)\geq c$ for all $t$), conditions, dependent on the monotonicity of the ratio $F(t)/f(t)$, can be found on $\pi$ in order that $\delta_{\pi}(X)$ is minimax.

The main feature of this paper is that, in the case where $F(t)/f(t)$ is nonincreasing, if $\pi(\|\theta\|^{2})$ is a superharmonic power prior of the form $\|\theta\|^{-2k}$ with $k>0$, the membership of the sampling density to the Berger class can be droped out. Also, our techniques are different from those in [5]. First, writing $\delta_{\pi}(X)=X+g(X)$ with $g(X)\propto \nabla M(\|X\|^{2})/m(\|X\|^{2})$, we follow Brandwein and Strawderman [4] proving that, for some $b>0$, the function $h=b\,\Delta M/m$ is subharmonic and satisfies $\|g\|^{2}/2\leq -h\leq -{\rm div}g$. Also, we adapt their approach using the fact that $R^{2(k+1)}\int_{B_{\theta,R}}h(x)\,dV_{\theta,R}(x)$ is nonincreasing in $R$ for any $\theta \in{\mathbb{R}} ^{p}$, when $V_{\theta,R}$ is the uniform distribution on the ball $B_{\theta,R}$ of radius $R$ and centered at $\theta$. Examples illustrate the theory.