Truncated linear estimation of a bounded multivariate normal mean

We consider the problem of estimating the mean θ of an N p ( θ , I p ) distribution with squared error loss ∥ δ − θ ∥ 2 and under the constraint ∥ θ ∥ ≤ m , for some constant m > 0 . Using Steinʹs identity to obtain unbiased estimates of risk, Karlinʹs sign change arguments, and conditional risk analysis, we compare the risk performance of truncated linear estimators with that of the maximum likelihood estimator δ mle . We obtain for fixed ( m , p ) sufficient conditions for dominance. An asymptotic framework is developed, where we demonstrate that the truncated linear minimax estimator dominates δ mle , and where we obtain simple and accurate measures of relative improvement in risk. Numerical evaluations illustrate the effectiveness of the asymptotic framework for approximating the risks for moderate or large values of p.