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, . . . , p, we consider estimating the vector θ = (θ1, . . . , θp) with loss ‖d− θ‖2 under the constraint ∑p i=1 (θi−τi) σ2 ≤ m2, with known τ1, . . . , τp, σ2,m. In comparing the risk performance of Bayesian estimators δα associated with uniform priors on spheres of radius α centered at (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