Prediction in Multivariate Mixed Linear Models with Equal Replications

The multivariate mixed linear model or multivariate components of variance model with equal replications is considered. The paper addresses the problem of predicting the sum of the regression mean and the random effects. When the feasible best linear unbiased predictors or empirical Bayes predictors are used, this prediction problem reduces to the estimation of the ratio of two covariance matrices. For the estimation of the ratio matrix, the James-Stein type estimators based on the Bartlett's decomposition, the Stein type orthogonally equivariant estimators and the Efron-Morris type estimators are obtained. Their dominance properties over the usual estimators including the unbiased one are established, and further domination results are shown by using information of order restriction between the two covariance matrices. It is also demonstrated that the empirical Bayes predictors that employs these improved estimators of the ratio of the two covariance matrices have uniformly smaller risks than the crude Efron-Morris type estimator in the context of estimation of a matrix mean in a fixed effects linear regression model where the components are unknown parameters. Finally, an extension of the model is given.