Restricted risk Bayes estimation for the mean of the multivariate normal distribution (Q1094018)

Let be \(X=(X_ 1,...,X_ p)^ t\) an observation from a p-variate normal distribution with unknwon 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 \(N_ p(\mu,A)\), where \(\mu\) and A are known. We find estimators of \(\theta\) which have small Bayes risk and which also satisfy the constraint \(R(\theta,\delta)\leq tr(Q\Sigma)+c\), \(R(\theta,\delta)\) being the most frequent risk of \(\delta\). Such estimates are good from both the most frequent and the Bayesian perspectives.