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

摘要

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 {norm of matrix} d - θ {norm of matrix}2 under the constraint ∑_i = 1~p frac((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 → ∞) frac(k (p), sqrt(p)) = sqrt(2), improving on the previously known sufficient condition of Marchand and Perron (2001) [3] for which k (p) ≥ sqrt(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 → ∞) frac(c_1 (p), sqrt(p)) = sqrt(frac(1, 3)).