**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

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.