Generalized Bayes Estimators in Multivariate Problems

Abstract

Several problems involving multivariate generalized Bayes estimators are investigated. First, a characterization of admissible estimators as generalized Bayes estimators is developed for certain multivariate exponential families and quadratic loss. The problem of verifying whether or not an estimator is generalized Bayes is also considered. Next, an important class of estimators for a multivariate normal mean is considered. (The class includes many minimax, empirical Bayes, and ridge regression estimators of current interest.) Necessary conditions are developed for an estimator in this class to be "nearly" generalized Bayes, in the sense that if it were properly smoothed, it would be generalized Bayes. An application to adaptive ridge regression is given. The paper concludes with the development of an asymptotic approximation to generalized Bayes estimators for general losses and location vector densities. Using this approximation, weakened versions of the above results are obtained for general losses and densities.