Estimation of a Common Mean and Recovery of Interblock Information

Abstract

Consider the problem of combining two unbiased estimators of a parameter when the estimators are known to be independent normal variables with unknown variances possibly unequal. The two one parameter families of estimators studied in Brown and Cohen, and Khatri and Shah, are accommodated in a single two parameter family studied in this paper and the results in the two papers are unified. For the type of estimators considered by Brown and Cohen, this paper not only offers a generalization but also a significant improvement. This improvement concerns the main result in Theorem 2.1 of Brown and Cohen and has bearing on their entire paper except the last section on interval estimation. Extensions of Brown and Cohen's Theorem 4.1 concerning the point estimation of the common mean of $K$-populations and Theorem 5.1 concerning interval estimation of the common mean of two populations are also presented.