Selecting a Minimax Estimator of a Multivariate Normal Mean

Abstract

The problem of estimating a $p$-variate normal mean under arbitrary quadratic loss when $p \geq 3$ is considered. Any estimator having uniformly smaller risk than the maximum likelihood estimator $\delta^0$ will have significantly smaller risk only in a fairly small region of the parameter space. A relatively simple minimax estimator is developed which allows the user to select the region in which significant improvement over $\delta^0$ is to be achieved. Since the desired region of improvement should probably be chosen to coincide with prior beliefs concerning the whereabouts of the normal mean, the estimator is also analyzed from a Bayesian viewpoint.