An approach to improving the Lindley estimator

Abstract Consider a p-variate (p 4) normal distribution with mean and identity covariancematrix. Using a simple property of noncentral chi square distribution, the generalizedBayes estimators dominating the Lindley estimator under quadratic loss are given basedon the methods of Brown, Brewster and Zidek for estimating a normal variance. Thisresult can be extended the cases where covariance matrix is completely unknown orP= ˙ 2 I for an unknown scalar ˙ 2 .Keywords: Generalized Bayes estimator, Lindley estimator, normal distribution,quadratic loss. 1. Introduction Let X = (X 1 ; ;X p ) 0 be a p-variate random vector normally distributed with unknownmean and the identity covariance matrix I. Then we consider the problem of estimatingby (X) relative to the quadratic loss function k(X) k 2 = ((X) ) 0 ((X) ).Every estimator will be evaluated by the risk function R(;(X)) = Ek(X) k 2 .Stein (1956) showed that the usual estimator X is inadmissible for p3 and Jamesand Stein (1961) constructed the improved estimator,