On estimating the mean In a bivariate normal distribution With equal or unequal variances

This paper considers the problem of estimating the mean μx of one of the components of the bivariate normal distribution with equal variances or unequal variances. When the mean of the other component μy is equal to μx it is advantageous to pool the two sample means as an estimator of μx. When the experimenter is uncertain whether μx = μy a preliminary test of significance is used at level α to test μx = μy. Three estimators of μx are considered, (i) preliminary test estimator (PTE), (ii) weighting function estimator (WFE), and (iii) adaptive preliminary test estimator (APTE). The WFE is defined as the linear combination of the two sample means with the weight obtained by minimizing the mean square error. The APTE is a PTE with the weight adopted from WFE. The biases, mean square errors, and relative efficiencies of all the three estimators are studied.