*J. Japan Statist. Soc.,* Vol. 33 (No. 2), pp. 245-270, 2003

## Prediction in Multivariate Mixed Linear Models

Tatsuya Kubokawa and Muni S. Srivastava

**Abstract. **
In the multivariate mixed linear model or multivariate components of variance model with equal replications, this 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. We propose scale equivariant shrinkage estimators for the ratio of the two covariance matrices. 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 employ 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 mean matrix in a fixed effects linear regression model where the components are unknown parameters.

*Key words and phrases*:
Decision theory, empirical Bayes procedure, multivariate components of variances, multivariate mixed linear model, posted land price data, shrinkage estimation, small area estimation.

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