Abstract: Based on data resampling techniques, two classes of empirical Bayes estimators are proposed for estimating the error variances in a heteroscedastic linear model. We concentrate primarily on the situation in which only a few replicates are available at each design point but the total number of observations n is relatively large. Properties of the empirical Bayes estimators, including invariance, robustness, consistency, asymptotic unbiasedness and mean squared error (MSE), are studied. In particular, a second order expansion of the MSE and an upper bound on the bias of the empirical Bayes estimator are given in terms of the diagonal elements of the projection matrix. Using these results, we compare the empirical Bayes estimator with other existing variance estimators. The MSE of the empirical Bayes estimator is smaller than that of the within-group sample variance and the MINQUE when n is large. Applications in inferences are also discussed. Some simulation results are presented.
Key words and phrases: Data resampling, empirical Bayes estimators, sample variance, MINQUE, consistency, bias, mean squared error, weighted least squares.