The Efficiency of Shrinkage Estimators with Respect to Zellner's Balanced Loss Function

Abstract

A setup with r uncorrelated linear models is considered. A generalization of Zellner's balanced loss function is proposed. Zellner's balanced loss function takes both error of estimation and goodness of fit into account. The classical loss function only considers error of estimation. Empirical Bayes and approximate minimum mean square error estimators are derived. The efficiency of these estimators is evaluated averaging over Zellner's balanced loss function. Comparisons are then made with the least square estimator and the analogous estimators for the classical loss function. Three kinds of James–Stein type estimators are considered that are similar to empirical Bayes estimators originally formulated by Rao, Wind and Dempster.