Improvement of the Liu estimator in linear regression model

Abstract

In the presence of stochastic prior information, in addition to the sample, Theil and Goldberger (1961) introduced a Mixed Estimator $\hat \beta _m$ for the parameter vector β in the standard multiple linear regression model (T,Xβ,σ² I). Recently, the Liu estimator which is an alternative biased estimator for β has been proposed by Liu (1993).

In this paper we introduce another new Liu type biased estimator called Stochastic restricted Liu estimator $\hat \beta _{srd}$ for β, and discuss its efficiency. The necessary and sufficient conditions for mean squared error matrix of the Stochastic restricted Liu estimator $\hat \beta _{srd}$ to exceed the mean squared error matrix of the mixed estimator $\hat \beta _m$ will be derived for the two cases in which the parametric restrictions are correct and are not correct. In particular we show that this new biased estimator is superior in the mean squared error matrix sense to both the Mixed estimator $\hat \beta _m$ and to the biased estimator introduced by Liu (1993).