Sankhya: The Indian Journal of Statistics

2004, Volume 66, Pt. 4,  678-706

Comparison of Bayesian and Frequentist Estimation and Prediction for a Normal Population

By

Cuirong Ren, South Dakota State University, Brookings, USA, Dongchu Sun, University of Missouri, Columbia, USA, Dipak K. Dey, University of Connecticut, Storrs, USA , CuirongRen, Dongchu Sun and Dipak K. Dey

SUMMARY. Comparisons of estimates between Bayes and frequentist methods are interesting and challenging topics in statistics. %Loss functions play a very important role in such comparisons. In this paper, Bayes estimates and predictors are derived for a normal distribution. The commonly used  requentist predictor such as the maximum likelihood estimate (MLE) is a plug-in" procedure by substituting the MLE of $\mu$ into the predictive distribution. We examine Bayes prediction under %various losses such as the $\alpha$-absolute error losses, the LINEX losses and the entropy loss as special case of the $\alpha$-absolute error losses. If the variance is unknown, the joint conjugate prior  Is used to estimate the unknown mean for the $\alpha$-absolute error losses and an {\it 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 estimate 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 naïve Bayesians who assume that Bayes methods always perform well regardless of circumstances.

AMS (1991) subject classification. 62F15, 62H10, 62H12 ;

Key words and phrases. Bayes estimation, Jeffreys prior, loss function, maximum likelihood estimator, risk function.

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