On efficient prediction and predictive density estimation for spherically symmetric models

Abstract

Let $X,U,Y$ be spherically symmetric distributed having density $${\eta }^{d+k/2}f\left(\eta (\Vert x-\theta {|}^2+\Vert u{\Vert }^2+\Vert y-c\theta {\Vert }^2)\right),$$ with unknown parameters $\theta \in {\mathbb{R}}^d$ and $\eta >0$, and with known density $f$ and constant $c>0$. Based on observing $X=x,U=u$, we consider the problem of obtaining a predictive density $\hat{q}(y;x,u)$ for $Y$ as measured by the expected Kullback-Leibler loss. A benchmark procedure is the minimum risk equivariant density ${\hat{q}}_{mre}$, which is Generalized Bayes with respect to the prior $\pi (\theta ,\eta )={\eta }^{-1}$. For $d\ge 3$, we obtain improvements on ${\hat{q}}_{mre}$, and further show that the dominance holds simultaneously for all $f$ subject to finite moments and finite risk conditions. We also obtain that the Bayes predictive density with respect to the harmonic prior ${\pi }_h(\theta ,\eta )={\eta }^{-1}\Vert \theta {\Vert }^{2-d}$ dominates ${\hat{q}}_{mre}$ simultaneously for all scale mixture of normals $f$.