On the estimation of means in a bivariate normal distribution with equal marginal variances

This paper considers the problem of estimation of the mean /1 of one of the components of a bivariate normal distribution with equal marginal variances from a sample of size n. The result of a preliminary test of hypothesis that the means /1 and /2 of the two components of the bivariate normal distribution are equal is used to define an estimator x̄* for /1. The bias and mean square error of this estimator are studied and the regions in the parameter space in which x̄* has smaller mean square error than x̄ the sample mean of the first component, are determined. The efficiency of this estimator relative to the usual estimator is tabulated and the tables can be used to determine a proper choice of significance level of the preliminary test.