Alternative Estimation of the Common Mean of Two Normal Populations with Order Restricted Variances

Abstract

The problem of estimating the common mean of two normal populations has been considered when it is known a priori that the variances are ordered. Under order restriction on the variances some new alternative estimators have been proposed including one that uses the maximum likelihood estimator (MLE) numerically. Further, it has been proved that each of these new estimators beats their unrestricted counterparts in terms of stochastic domination as well as Pitman measure of closeness criterion. Sufficient conditions for improving estimators in certain classes of equivariant estimators have been proved, and consequently improved estimators have been obtained under order restriction on the variances. A detailed simulation study has been done in order to evaluate the performances of all the proposed estimators using an existing estimator as a benchmark. From our simulation study, it has been established that the new alternative estimators improve significantly upon their unrestricted counterparts and compete well with an existing estimator under order restriction on the variances.