Liu-type Shrinkage Estimations in Linear Models

Abstract

In this study, we present the preliminary test, Stein-type and positive part Liu estimators in the linear models when the parameter vector $\mathbf{\beta }$ is partitioned into two parts, namely, the main effects ${\mathbf{\beta }}_1$ and the nuisance effects ${\mathbf{\beta }}_2$ such that $\mathbf{\beta }=\left({\mathbf{\beta }}_1,{\mathbf{\beta }}_2\right)$. We consider the case that a priori known or suspected set of the explanatory variables do not contribute to predict the response so that a sub-model may be enough for this purpose. Thus, the main interest is to estimate ${\mathbf{\beta }}_1$ when ${\mathbf{\beta }}_2$ is close to zero. Therefore, we conduct a Monte Carlo simulation study to evaluate the relative efficiency of the suggested estimators, where we demonstrate the superiority of the proposed estimators.