Empirical Bayes Estimation in Cross-Classified Gaussian Models With Unbalanced Design

The James-Stein estimator and its Bayesian interpretation demonstrated the usefulness of empirical Bayes methods in facilitating competitive shrinkage estimators for multivariate problems consisting of nonrandom parameters. When transitioning from homoscedastic to heteroscedastic Gaussian data, empirical ``linear Bayes estimators typically lose attractive properties such as minimaxity, and are usually justified mainly from Bayesian viewpoints. Nevertheless, by appealing to frequentist considerations, traditional empirical linear Bayes estimators can be modified to better accommodate the asymmetry in unequal variance cases. This work develops empirical Bayes estimators for cross-classified (factorial) data with unbalanced design that are asymptotically optimal within classes of shrinkage estimators, and in particular asymptotically dominate traditional parametric empirical Bayes estimators as well the usual (unbiased) estimator