Liu-type Shrinkage Estimations in Linear Models

In this study, we present the preliminary test, Stein-type and positive part Liu estimators in the linear models when the parameter vector $\boldsymbol{\beta}$ is partitioned into two parts, namely, the main effects $\boldsymbol{\beta}_1$ and the nuisance effects $\boldsymbol{\beta}_2$ such that $\boldsymbol{\beta}=\left(\boldsymbol{\beta}_1, \boldsymbol{\beta}_2 \right)$. 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 $\boldsymbol{\beta}_1$ when $\boldsymbol{\beta}_2$ 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.