On efficient prediction and predictive density estimation for spherically symmetric models

TL;DR

This paper develops improved predictive density estimators for spherically symmetric models, demonstrating dominance over traditional methods across various distributions and conditions, especially for dimensions three and higher.

Contribution

It introduces new predictive density estimators that outperform the minimum risk equivariant estimator in spherically symmetric models, applicable to a broad class of distributions.

Findings

Improved predictive densities dominate the benchmark for dimensions d ≥ 3.

Dominance holds across all distributions with finite moments and risk.

Bayesian predictive densities with harmonic priors outperform traditional methods.

Abstract

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 $f$.