General Classes of Shrinkage Estimators for the Multivariate Normal Mean with Unknown Variance: Minimaxity and Limit of Risks Ratios

Abdelkader Benkhaled ; Abdenour Hamdaoui

Kragujevac Journal of Mathematics, Tome 46 (2022) no. 2, p. 193

Résumé

In this paper, we consider two forms of shrinkage estimators of the mean $\theta$ of a multivariate normal distribution $X \sim $ $N_{p}\left(\theta, \sigma^{2}I_{p}\right)$ in $\mathbb{R}^{p}$ where $\sigma^{2}$ is unknown and estimated by the statistic $S^{2}$ ($S^{2}\sim \sigma^{2}\chi_{n}^{2}$). Estimators that shrink the components of the usual estimator $X$ to zero and estimators of Lindley-type, that shrink the components of the usual estimator to the random variable $\overline{X}.$ Our aim is to improve under appropriate condition the results related to risks ratios of shrinkage estimators, when $n$ and $p$ tend to infinity and to ameliorate the results of minimaxity obtained previously of estimators cited above, when the dimension $p$ is finite. Some numerical results are also provided.

Détail

Classification : 62F12 62C20
Keywords: James-Stein estimator, multivariate Gaussian random variable, non-central chi-square distribution, quadratic risk, shrinkage estimator