Restricted risk Bayes estimation for the mean of the multivariate normal distribution

Let X = (X1,...,Xp)t to be an observation from a p-variate normal distribution with unknown mean vector [theta] = ([theta]1,...,[theta]p)t and known covariance matrix [Sigma]. It is desired to estimate [theta] under the quadratic loss L([theta], [delta]) = ([theta] - [delta])tQ([theta] - [delta]). Suppose prior beliefs concerning [theta] can be approximately modeled by a conjugate prior distribution [pi] which is Np([mu], A), where [mu] and A are known. We find estimators of [theta] which have small Bayes risk and which also satisfy the constraint R([theta], [delta]) <= tr(Q [Sigma]) + c, R([theta], [delta]) being the most frequent risk of [delta]. Such estimates are good from both the most frequent and the Bayesian perspectives.