On efficient prediction and predictive density estimation for spherically symmetric models

Let $X,U,Y$ be spherically symmetric distributed having density $\eta^{d +k/2} \, f\left(\eta(\|x-\theta|^2+ \|u\|^2 + \|y-c\theta\|^2 ) \right)\,,$ 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 $d \geq 3$, 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 $\pi_h(\theta, \eta) =\eta^{-1} \|\theta\|^{2-d}$ dominates $\hat{q}_{mre}$ simultaneously for all scale mixture of normals $f$.