A Bayesian Approach to the Linear Model with Unequal Variances

Abstract

Estimation procedures are proposed for the means and variances of several normal distributions. They are valid in cases where the parameters of different distributions may be unequal but are thought a priori to be related in certain ways. A log-linear model is assumed for the variances, together with a linear model for the means. Some general Bayesian results are obtained, and some special cases are discussed including the important situation where parameters of different distributions are a priori exchangeable. The posterior estimates then adjust the standard estimates, by shrinking them towards central values based on collateral information. A numerical example concerns the simultaneous estimation of the variances of the observed breaking strengths of six fabrics.

KEY WORDS:

• Linear models
• Heteroscedastic variances
• Log-linear models
• Multivariate normal prior distributions
• Posterior means and modes
• Exchangeability
• Shrinkage of estimates
• Partial pooling processes
• Bartlett's test
• Regression models
• Two-way Analysis of Variance
• Autoregressive Processes