Abstract: The Lasso is an attractive approach to variable selection in sparse, high-dimensional regression models. Much work has been done to study the selection and estimation properties of the Lasso in the context of least squares regression. However, the least squares based method is sensitive to outliers. An alternative to the least squares method is the least absolute deviations (LAD) method which is robust to outliers in the responses. In this paper, we study the selection and estimation properties of the Lasso in LAD regression. We provide sufficient conditions under which the LAD-Lasso is estimation or selection consistent in sparse, high-dimensional settings. We use simulation studies to evaluate the performance of the LAD-Lasso, and compare the proposed method with the LS-Lasso in a range of generating models.
Key words and phrases: Consistency, high-dimensional model, Lasso, robust regression, sparsity, variable selection.