Minimaxity and Limits of Risks Ratios of Shrinkage Estimators of a Multivariate Normal Mean in the Bayesian Case

Abstract

In this article, we consider two forms of shrinkage estimators of the mean $\theta$ of a multivariate normal distribution $X\sim N_p\left(\theta ,{\sigma }^2{I}_p\right)$ where ${\sigma }^2$ is unknown. We take the prior law $\theta \sim N_p\left(\upsilon ,{\tau }^2{I}_p\right)$ and we constuct a Modified Bayes estimator ${\delta }_B^{\ast }$ and an Empirical Modified Bayes estimator ${\delta }_{EB}^{\ast }$. We are interested in studying the minimaxity and the limits of risks ratios of these estimators, to the maximum likelihood estimator $X$, when $n$ and $p$ tend to infinity.

arXiv comment17 pages, 6 figures