Asymptotic properties of risks ratios of shrinkage estimators

Abstract

We study the estimation of the mean of a multivariate normal distribution Np ; Ip in R; is unknown and estimated by the chi-square variable S 2 2n: In this work we are interested in studying bounds and limits of risk ratios of shrinkage estimators to the maximum likelihood estimator, when n and p tend to in nity provided that limp!1 k k p 2 = c:The risk ratio for this class of estimators has a lower bound Bm = c 1 + c ; when n and p tend to in nity provided that limp!1 k k p 2 = c:We give simple conditions for shrinkage minimax estimators, to attain the limiting lower bound Bm:We also show that the risk ratio of James-Stein estimator and those that dominate it, attain this lower bound Bm (in particularly its positive-part version).We graph the corresponding risk ratios for estimators of James-Stein JS ; its positive part +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. Keywords : James-Stein estimator, multivariate gaussian random variable, non-central chi-square distribution, shrinkage estimator, quadratic risk. 2000 AMS Classi cation: 62F12, 62C20