Asymptotic properties of risks ratios of shrinkage estimators

Abstract

We study the estimation of the mean $\theta$ of a multivariate normal distribution $N_p(\theta,\sigma^2I_p)$ in $\mathbb{R}^p$, $\sigma^2$ is unknown and estimated by the chi-square variable $S^2\sim \sigma^2\chi_n^2$. In this work we are interested in studying bounds and limits of risk ratios of shrinkage estimators to the maximum likelihood estimators, when $n$ and $p$ tend to infinity provided that $\lim_{p\to\infty}\dfrac{\|\theta\|^2}{p\sigma^2}=c$. We give simple conditions for shrinkage minimax estimators, to attain the limiting lower bound $B_m$. We also show that the risk ratio of James-Stein estimator and those that dominate it, attain this lower bound $B_m$ (in particularly its positive-part version). We graph the corresponding risk ratios for estimators of James-Stein $\delta_{JS}$, its positive part $\delta_{JS}^+$, that of a minimax estimator, and an estimator dominating the James-Stein estimator in the sense of the quadratic risk (polynomial estimators proposed by Tze Fen Li and Hou Wen Kuo [13]) for some values of $n$ and $p$.