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(θi,σ2),i=1,…,pXi∼N(θi,σ2),i=1,…,p, we consider estimating the vector θ=(θ1,…,θp)′θ=(θ1,…,θp)′ with loss ‖d−θ‖2‖d−θ‖2 under the constraint ∑i=1p(θi−τi)2σ2≤m2, with known τ1,…,τp,σ2,mτ1,…,τp,σ2,m. In comparing the risk performance of Bayesian estimators δαδα associated with uniform priors on spheres of radius αα centered at (τ1,…,τp)(τ1,…,τp) with that of the maximum likelihood estimator δmle, 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)(m,p) necessary and sufficient conditions on αα for δαδα to dominate δmle. Large sample determinations of these conditions are provided. Both cases where all such δαδα’s and cases where no such δαδα’s dominate δmle are elicited. We establish, as a particular case, that the boundary uniform Bayes estimator δmδm dominates δmle if and only if m≤k(p)m≤k(p) with limp→∞k(p)p=2, improving on the previously known sufficient condition of Marchand and Perron (2001) [3] for which k(p)≥p. Finally, we improve upon a universal dominance condition due to Marchand and Perron, by establishing that all   Bayesian estimators δπδπ with ππ spherically symmetric and supported on the parameter space dominate δmle whenever m≤c1(p)m≤c1(p) with limp→∞c1(p)p=13.