Optimal Estimation of Multidimensional Normal Means With an Unknown Variance

Let X ∼ Np(θ, σ2 Ip) and W ∼ σ2 χ2m, where both θ and σ2 are unknown, and X is independent of W. Optimal estimation of θ with unknown σ2 is a fundamental issue in applications but basic theoretical issues remain open. We consider estimation of θ under squared error loss. We develop sufficient conditions for prior density functions such that the corresponding generalized Bayes estimators for θ are admissible. This paper has a two-fold purpose: 1. Provide a benchmark for the evaluation of shrinkage estimation for a multivariate normal mean with an unknown variance; 2. Use admissibility as a criterion to select priors for hierarchical Bayes models. To illustrate how to select hierarchical priors, we apply these sufficient conditions to a widely used hierarchical Bayes model proposed by Maruyama & Strawderman [M-S] (2005), and obtain a class of admissible and minimax generalized Bayes estimators for the normal mean θ. We also develop necessary conditions for admissibility of generalized Bayes estimators in the M-S (2005) hierarchical Bayes model. All the results in this paper can be directly applied in the familiar setting of Gaussian linear regression.