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 ∼ N ( θ i , σ 2 ) , i = 1 , … , p , we consider estimating the vector θ = ( θ 1 , … , θ p ) ′ with loss ‖ d − θ ‖ 2 under the constraint ∑ i = 1 p ( θ i − τ i ) 2 σ 2 ≤ m 2 , with known τ 1 , … , τ p , σ 2 , m . In comparing the risk performance of Bayesian estimators δ α associated with uniform priors on spheres of radius α centered at ( τ 1 , … , τ p ) with that of the maximum likelihood estimator δ 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 α for δ α to dominate δ mle . Large sample determinations of these conditions are provided. Both cases where all such δ α ’s and cases where no such δ α ’s dominate δ mle are elicited. We establish, as a particular case, that the boundary uniform Bayes estimator δ m dominates δ mle if and only if m ≤ k ( p ) with lim p → ∞ 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 δ π with π spherically symmetric and supported on the parameter space dominate δ mle whenever m ≤ c 1 ( p ) with lim p → ∞ c 1 ( p ) p = 1 3 .