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 weighted L1β-penalty, called weighted robust Lasso
(WR-Lasso), in which weights are introduced to ameliorate the bias problem
induced by the L1β-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 the L1β-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.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1191 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org