Shrinkage and pretest Liu estimators in semiparametric linear measurement error models

In this article for semiparametric linear mesurement errors models under a multicollinearity setting, we define five shrinkage Liu estimators, namely, ordinary Liu estimator, restricted Liu estimator (RLE), preliminary test Liu estimator (Ple), Stien Liu estimator (Sle) and positive stein Liu estimator (Psle) for estimating the parameters when it is suspected that the parameter beta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document} may belong to a linear subspace defined by H beta=c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\textbf{H}}\beta = c$$\end{document}. Asymptotic properties of the estimators are studied with respect to quadratic risks. We derive the biases and quadratic risk expressions of these estimators and obtain the region of optimality of each estimator. Also, necessary and sufficient conditions, for the superiority of the shrinkage Liu estimator over its counterpart, for choosing the Liu parameter d are established. Finally, we illustrate the performance of the proposed shrinkage estimators with a simulation study and real data analyses.