Estimation of multivariate normal mean and its application to mixed linear models

Abstract

<p>Let X = (x(,1),x(,2),...,x(,p))' be a multivariate normal random variable with mean vector, (theta), in a space (THETA), and variance matrix I;From Strawderman's (1971) class of estimators, we derive a minimax admissible estimator for (theta). It has a relatively simple form when p is greater than or equal to five. We also extend Stein's (1973) technique to evaluate unbiased estimators of risks for discontinuous estimators. Then, we show the exact risks of a preliminary test estimator and of compromised or mixture estimators. We develop estimators that shrink towards some subspace of (THETA) and show the relationship between shrinkage functionals and variance component estimators in balanced mixed linear models. We also investigate the asymptotic behavior of shrinkage estimators. By choosing an appropriate subspace, we show that our estimator and ridge regression estimators achieve stability of prediction in a particular data example;References;Strawderman, W. E. 1971. Proper Bayes Minimax Estimators of the Multivariate Normal Mean. The Annals of Mathematical Statistics 42:385-388. Stein, C. 1973. Estimation of the Mean of a Multivariate Distribution Proceedings of the Prague Symposium on Asymptotic Statistics:345-387.</p>