Bayesian predictive densities for linear regression models under alpha-divergence loss: some results and open problems

TL;DR

This paper derives Bayesian predictive densities for normal linear models under alpha-divergence loss, providing explicit formulas and dominance results for certain hierarchical priors, and discusses open problems in the area.

Contribution

It introduces a general canonical form for predictive density estimation under alpha-divergence loss and derives explicit generalized Bayes solutions for specific hierarchical priors.

Findings

Derived explicit generalized Bayes predictive densities for alpha-divergence loss.

Established dominance of certain estimators over invariant prior-based estimators for alpha=1.

Identified open problems and future research directions in predictive density estimation.

Abstract

This paper considers estimation of the predictive density for a normal linear model with unknown variance under alpha-divergence loss for -1 <= alpha <= 1. We first give a general canonical form for the problem, and then give general expressions for the generalized Bayes solution under the above loss for each alpha. For a particular class of hierarchical generalized priors studied in Maruyama and Strawderman (2005, 2006) for the problems of estimating the mean vector and the variance respectively, we give the generalized Bayes predictive density. Additionally, we show that, for a subclass of these priors, the resulting estimator dominates the generalized Bayes estimator with respect to the right invariant prior when alpha=1, i.e., the best (fully) equivariant minimax estimator.