Variable selection in high-dimensional extremile regression via the quasi elastic net

Abstract

Extremile regression proposed in recent years not only retains the advantage of quantile regression that can fully show the information of sample data by setting different quantiles, but also has its own superiority compared with quantile regression and expectile regression, due to its explicit expression and conservativeness in estimating. Here, we propose a linear extremile regression model and introduce a variable selection method using a penalty called a quasi elastic net (QEN) to solve high-dimensional problems. Moreover, we propose an EM algorithm and establish corresponding theoretical properties under some mild conditions. In numerical studies, we compare the QEN penalty with the <inline-formula> <tex-math id="M1">\begin{document}$L_{0}$\end{document}</tex-math> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0099_M1.jpg"/> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0099_M1.png"/> </alternatives> </inline-formula>, <inline-formula> <tex-math id="M2">\begin{document}$L_{1}$\end{document}</tex-math> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0099_M2.jpg"/> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0099_M2.png"/> </alternatives> </inline-formula>, <inline-formula> <tex-math id="M3">\begin{document}$L_{2}$\end{document}</tex-math> <alternatives> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0099_M3.jpg"/> <graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JUSTC-2022-0099_M3.png"/> </alternatives> </inline-formula> and elastic net penalties, and the results show that the proposed method is effective and has certain advantages in analysis.