Bayesian Methods for Variance Component Models

Abstract

An exact Bayesian analysis can be performed for normal theory variance components models, using importance sampling. But remarkably accurate approximations are available using the Laplacian T-approximation introduced by Leonard, Hsu, and Ritter. Instead of maximizing the joint posterior density, conditional upon the parameter of interest, a device described by O'Hagan is used first. The Bayesian estimators are compared to the Lindley—Stein shrinkage estimators and the Lindley—Smith joint modal estimators. It is confirmed that joint modes can overcollapse toward prior hypotheses, when compared with more sensible Bayesian procedures. This is referred to as a “collapsing phenomenon.” A numerical example from the one-way random-effects model is considered, and the risks of the different estimators are simulated under a variety of loss functions. It is concluded that although the Lindley—Stein estimator performs well, a full hierarchical Bayesian analysis performs at least equally well, while permitting more detailed finite-sample inference regarding any parameter of interest.

Key Words:

• Analysis of variance
• Empirical Bayes
• Hierarchical Bayes
• Importance sampling
• Joint mode
• Laplacian T approximation
• Marginal mode
• Maximum component loss
• Risk function
• Shrinkage estimator