Revisiting a General Pattern of Domination Using the Graybill–Deal Estimator

This is a review article in a popular area of research which Professor J.N.K. Rao had touched upon during his distinguished research career. We convey our congratulations and best wishes to JNK Rao on his attaining 80 yrs. We wish him a long, active and peaceful life in the years to come. Herein we review the problem of unbiased estimation of the common parameter (θ) involved in the linear regression models of the means of two independent normal populations with unequal and unknown variances. We examine the popular Graybill–Deal estimator for the common parameter θ and ask the question: When will the Graybill–Deal estimator possess uniformly smaller variance than that of the individual unbiased estimators θ̂1 and θ̂2 arising out of the two given models? It turns out that the result depends only on the quantities m1 and m2, corresponding to the error degrees of freedom, irrespective of the nature of the linear mean models. We find the same result continues to hold in situations wherein the p (p ≥ 2) linear regression models involve k (k > 1) common estimable parameter(s) in the mean models. In this context, we use the criterion of ‘Loewner Order Domination’ of information or dispersion matrices.