Bayes, Minimax and Nonnegative Estimators of Variance Components under Kullback-Leibler Loss

In a balanced one-way model with random effects, the simultaneous estimation of the variance components are considered under the intrinsic Kullback-Leibler loss function. The uniformly minimum variance unbiased (UMVU) or ANOVA estimators are known to have a drawback of taking negative values. The paper shows the minimaxity of the ANOVA estimators of the variance components and obtains classes of minimax estimators. Out of these classes, two types of minimax and nonnegative estimators are singled out, and they are characterized as empirical Bayes and generalized Bayes estimators. Also a residual maximum likelihood (REML) estimator is interpreted as an empirical Bayes rule. The risk performances of the derived estimators are investigated based on simulation experiments. An extension to the general mixed linear model with two components of variances is studied, and nonnegative estimators improving on the ANOVA estimators are given.