James-Stein estimators for the mean vector of a multivariate normal population based on independent samples from two normal populations with common covariance structure

Abstract

The paper considers shrinkage estimators of the mean vector of a multivariate normal population based on independent random samples from two multivariate normal populations with different mean vectors but common covariance structure. The shrinkage and the positive-rule shrinkage estimators are defined by using the preliminary test approach when uncertain prior information regarding the equality of the two population mean vectors is available. The properties and performances of the estimators are investigated. The performances of the estimators are compared based on the unbiasedness and quadratic risk criteria. The relative performances of the estimators are discussed under different conditions. The shrinkage estimator dominates the maximum likelihood estimator, and the positive-rule shrinkage estimator uniformly over performs the shrinkage estimator with respect to the quadratic risk.