c © 2004, Indian Statistical Institute Comparison of Bayesian and Frequentist Estimation and Prediction for a Normal Population

Comparisons of estimates between Bayes and frequentist methods are inter-esting and challenging topics in statistics. In this paper, Bayes estimates and predictors are derived for a normal distribution. The commonly used frequentist predictor such as the maximum likelihood estimate (MLE) is a “plug-in ” procedure by substituting the MLE of µ into the predictive dis-tribution. We examine Bayes prediction under the α-absolute error losses, the LINEX losses and the entropy loss as special case of the α-absolute er-ror losses. If the variance is unknown, the joint conjugate prior is used to estimate the unknown mean for the α-absolute error losses and an ad hoc method by replacing the unknown variance by the sample variance for the LINEX losses. Bayes estimates are also extended to the linear combinations of regression coefficients. Under certain assumptions for a design matrix, the asymptotic expected losses are derived. Under suitable priors, Bayes esti-mate and predictor perform better than the MLE. Under the LINEX loss, the Bayes estimate under the Jeffreys prior is superior to the MLE. However, for prediction, it is not clear whether Bayes prediction or MLE performs better. Under some circumstances, even when one loss is the “true ” loss function, Bayes estimate under another loss performs better than the Bayes estimate under the “true ” loss. This serves as a warning to naive Bayesians who assume that Bayes methods always perform well regardless of circumstances