A bayes method for estimating the common mean of two normal populations

Abstract

Suppose we wish to estimate the common mean μof two normal samples with unknown variances .If are equal, we may average the two sample estimators of μindependently of the variances. This procedure is known to be admissible in the sense that no other estimator has uniformly smaller mean squared error. In the case when are not equal, many combination procedures have been proposed. The maximum likelihood procedure is also known to be admissible in a restricted class of estimators. A procedure proposed by Zacks combines these two admissible procedures based on the result of a preliminary test for the equality of . We propose Bayes estimators for μderived from carefully chosen prior distributions and give simulation results to show that the Zacks estimators involving a preliminary test are uniformly beaten by the Bayes procedures.