Ultrahigh dimensional variable selection through the penalized maximum trimmed likelihood estimator

The penalized maximum likelihood estimator (PMLE) has been widely used for variable selection in high-dimensional data. Various penalty functions have been employed for this purpose, e.g., Lasso, weighted Lasso, or smoothly clipped absolute deviations. However, the PMLE can be very sensitive to outliers in the data, especially to outliers in the covariates (leverage points). In order to overcome this disadvantage, the usage of the penalized maximum trimmed likelihood estimator (PMTLE) is proposed to estimate the unknown parameters in a robust way. The computation of the PMTLE takes advantage of the same technology as used for PMLE but here the estimation is based on subsamples only. The breakdown point properties of the PMTLE are discussed using the notion of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d$$\end{document}-fullness. The performance of the proposed estimator is evaluated in a simulation study for the classical multiple linear and Poisson linear regression models.