Estimation of heterogeneous variances using empirical Bayes methods: theoretical considerations

Procedures are described to estimate variances when heterogeneity of genetic and residual dispersion parameters exists for some criterion. Genetic and residual variances are considered to follow distributions with either known or unknown parameters. The estimates of variances obtained are weighted averages of the corresponding parameter and of a data-based statistic. Although the techniques presented are largely inspired by Bayesian ideas, the procedures can be given a frequentist interpretation, and the parameters of the prior distributions can be estimated from the data at hand. Techniques are described and illustrated for situations in which animals are related or unrelated across herds. We conjecture that the proposed estimators have smaller mean squared error than those obtained by grouping observations in some way and then applying REML within each group.