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: X-i similar to N(theta(i), sigma(2)), i= 1.....p, we consider estimating the vector theta = (theta(1),...,theta(p))' with loss parallel to d - theta parallel to(2) under the constraint Sigma(p)(i=1) (o(i)-t(i))(2)/sigma(2) <= m(2), with known t(1),...,t(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 (t(1),...,t(p)) with that of the maximum likelihood estimator delta(mle), 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 delta(mle). 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 delta(mle) are elicited. We establish, as a particular case, that the boundary uniform Bayes estimator delta(m) dominates delta(mle) if and only if m <= k(p) with lim(p ->infinity) k(p)/root p = root 2, improving on the previously known sufficient condition of Marchand and Perron (2001) [3] for which k(p) >= root p. 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 delta(mle) whenever m <= c(1)(p) with lim(p ->infinity) c1(p)/root p = root 1/3. (C) 2010 Elsevier Inc. All rights reserved.