Variable Selection in Quantile Regression

After its inception in Koenker and Bassett (1978), quantile regression has become an important and widely used technique to study the whole conditional distribution of a response variable and grown into an important tool of applied statistics over the last three decades. In this work, we focus on the variable selection aspect of penalized quantile regression. Under some mild conditions, we demonstrate the oracle properties of the SCAD and adaptive-LASSO penalized quantile regressions. For the SCAD penalty, despite its good asymptotic properties, the corresponding optimization problem is non-convex and as a result much harder to solve. In this work, we take advantage of the decomposition of the SCAD penalty function as the difference of two convex functions and propose to solve the corresponding optimization using the Difference Convex Algorithm (DCA).