On the Asymptotic Efficiency of Least Squares Estimators

Abstract

The problem of estimating a linear transformation between two finite dimensional vector spaces is considered, when the observed vectors in both spaces are subject to error, and there is an indefinitely increasing number of replications of a fixed number of treatments. A general class of ordinary estimators is defined, and it is shown that, in the case of homogeneity of variances, the simple least squares estimators are asymptotically efficient within the class of ordinary estimators, in the sense that they minimize, within that class, the asymptotic mean square error of prediction. When the error variances are unequal, however, the asymptotically efficient estimators are weighted least squares estimators, whose weights are based on preliminary estimators of the linear transformation and the error variances.