Asymptotic analysis of high dimensional LAD regression with Lasso
The Lasso is an attractive approach to variable selection in sparse, highdimensional 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.