Date of this Version
The Annals of Statistics
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.
normal mean problem, admissibility, generalized Bayes estimator, unknown variance, shrinkage estimator, minimaxity
Brown, L., & Han, X. (2011). Optimal Estimation of Multidimensional Normal Means With an Unknown Variance. The Annals of Statistics, 1-31. Retrieved from https://repository.upenn.edu/statistics_papers/43
Date Posted: 27 November 2017
This document has been peer reviewed.