On admissible estimation of a mean vector when the scale is unknown

Abstract

We consider admissibility of generalized Bayes estimators of the mean of a p-variate normal distribution when the scale is unknown, and the loss is quadratic. The priors considered put the improper invariant prior on the scale while the prior on the mean has a hierarchical normal structure conditional on the scale. This conditional hierarchical prior is indexed by a hyperparameter, a. In earlier studies, the authors established admissibility/inadmissibility of the generalized Bayes estimator under the proper/improper conditional prior ($a>-1$ / $a<-2$), respectively. In this paper we complete the admissibility/inadmissibility characterization for this class of priors by establishing admissibility for the improper conditional prior ($-2\le a\le -1$). This boundary, $a=-2$, with admissibility for $a\ge -2$ and inadmissibility for $a<-2$ corresponds exactly to that in the known scale case for this class of conditional priors, and which follows from Brown’s 1971 paper. As a notable benefit of this enlargement of the class of admissible generalized Bayes estimators, we give admissible and minimax estimators for $p\ge 3$ as opposed to an earlier study which required $p\ge 5$. In one particularly interesting special case, we establish that the joint Stein prior for the unknown scale case leads to a minimax admissible estimator for $p\ge 3$.