Admissible Predictive Density Estimation

Let $X|\mu \sim N_{p}(\mu ,v_{x}I)$ and $Y|\mu \sim N_{p}(\mu ,v_{y}I)$ be independent p-dimensional multivariate normal vectors with common unknown mean .. Based on observing X = x, we consider the problem of estimating the true predictive density p(y|.) 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.