Optimal Estimation of Multidimensional Normal Means with Unknown Variances

• Activity Number: 519
• Type: Contributed
• Date/Time: Wednesday, August 4, 2010 : 10:30 AM to 12:20 PM
• Sponsor: IMS
• Author(s): Xu Han*+ and Lawrence D. Brown
• Companies: Princeton University and University of Pennsylvania
• Keywords: normal mean problem ; admissibility ; generalized Bayes estimator ; unknown variance ; shrinkage estimator ; minimaxity

Abstract

Let $X\sim N_p(\theta,\sigma^2I)$ and $W\sim\sigma^2\chi_m^2$, where both $\theta$ and $\sigma^2$ are unknown. We consider estimation of $\theta$ under squared error loss. We develop sufficient conditions for prior density functions such that the corresponding generalized Bayes estimators for $\theta$ 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 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 (2005), and obtain a class of admissible and minimax generalized Bayes estimators for the normal mean $\theta$.