title: ADAPTIVE ROBUST VARIABLE SELECTION.
abstract: Heavy-tailed high-dimensional data are commonly encountered in various scientific fields and pose great challenges to modern statistical analysis. A natural procedure to address this problem is to use penalized quantile regression with weightedL1-penalty, called weighted robust Lasso (WR-Lasso), in which weights are introduced to ameliorate the bias problem induced by theL1-penalty. In the ultra-high dimensional setting, where the dimensionality can grow exponentially with the sample size, we investigate the model selection oracle property and establish the asymptotic normality of the WR-Lasso. We show that only mild conditions on the model error distribution are needed. Our theoretical results also reveal that adaptive choice of the weight vector is essential for the WR-Lasso to enjoy these nice asymptotic properties. To make the WR-Lasso practically feasible, we propose a two-step procedure, called adaptive robust Lasso (AR-Lasso), in which the weight vector in the second step is constructed based on theL1-penalized quantile regression estimate from the first step. This two-step procedure is justified theoretically to possess the oracle property and the asymptotic normality. Numerical studies demonstrate the favorable finite-sample performance of the AR-Lasso.
authors.affiliation: Princeton University, University of Southern California and IBM T.J. Watson Research Center.
authors.forename: Jianqing
authors.initials: J
authors.lastname: Fan
authors.affiliation: Princeton University, University of Southern California and IBM T.J. Watson Research Center.
authors.forename: Yingying
authors.initials: Y
authors.lastname: Fan
authors.affiliation: Princeton University, University of Southern California and IBM T.J. Watson Research Center.
authors.forename: Emre
authors.initials: E
authors.lastname: Barut
keywords: Adaptive weighted L1; High dimensions; Oracle properties; Robust regularization