Admissible predictive density estimation

TL;DR

This paper characterizes admissible predictive density estimators for multivariate normal models under Kullback-Leibler loss, showing that all generalized Bayes rules form a complete class and providing conditions for their admissibility.

Contribution

It establishes that the class of all generalized Bayes rules is complete and identifies conditions under which formal Bayes rules are admissible in predictive density estimation.

Findings

Generalized Bayes rules form a complete class.

Conditions from Brown and Hwang ensure admissibility.

Provides a framework for admissible predictive density estimation.

Abstract

Let $X|\mu\sim N_p(\mu,v_xI)$ and $Y|\mu\sim N_p(\mu,v_yI)$ be independent $p$-dimensional multivariate normal vectors with common unknown mean $\mu$. Based on observing $X=x$, we consider the problem of estimating the true predictive density $p(y|\mu)$ of $Y$ under expected Kullback--Leibler loss. Our focus here is the characterization of admissible procedures for this problem. We show that the class of all generalized Bayes rules is a complete class, and that the easily interpretable conditions of Brown and Hwang [Statistical Decision Theory and Related Topics (1982) III 205--230] are sufficient for a formal Bayes rule to be admissible.