On Bayes estimators with uniform priors on spheres and their comparative performance with maximum likelihood estimators for estimating bounded multivariate normal means

For independently distributed observables: Xi~N([theta]i,[sigma]2),i=1,...,p, we consider estimating the vector [theta]=([theta]1,...,[theta]p)' with loss ||d-[theta]||2 under the constraint , with known [tau]1,...,[tau]p,[sigma]2,m. In comparing the risk performance of Bayesian estimators [delta][alpha] associated with uniform priors on spheres of radius [alpha] centered at ([tau]1,...,[tau]p) with that of the maximum likelihood estimator , we make use of Stein's unbiased estimate of risk technique, Karlin's sign change arguments, and a conditional risk analysis to obtain for a fixed (m,p) necessary and sufficient conditions on [alpha] for [delta][alpha] to dominate . Large sample determinations of these conditions are provided. Both cases where all such [delta][alpha]'s and cases where no such [delta][alpha]'s dominate are elicited. We establish, as a particular case, that the boundary uniform Bayes estimator [delta]m dominates if and only if m<=k(p) with , improving on the previously known sufficient condition of Marchand and Perron (2001) [3] for which . Finally, we improve upon a universal dominance condition due to Marchand and Perron, by establishing that all Bayesian estimators [delta][pi] with [pi] spherically symmetric and supported on the parameter space dominate whenever m<=c1(p) with .