Variable selection for ultra-high dimensional quantile regression with Huber approximation

To address the challenge of variable selection in ultra-high dimensional quantile regression, we introduce a coordinate descent algorithm that incorporates adaptive Lasso (ALasso), SCAD and MCP penalties using the Huber approximation method. The proposed method reformulates the quantile regression problem into a more tractable optimization framework and uses penalty terms to identify significant variables. For non-convex penalties (SCAD and MCP), we adopt a local linear approximation (LLA) strategy that iteratively approximates the non-convex penalty by a weighted Lasso problem, enabling efficient coordinate-wise updates with closed-form solutions. The convergence properties of the algorithm are established. To further improve selection accuracy, we propose a two-step procedure that combines quantile correlation-based sure independence screening (QCSIS) with the penalized coordinate descent algorithm. In the first step, QCSIS reduces dimensionality by screening the ultra-high dimensional variables. In the second step, the penalized algorithm refines the selection from the screened subset. Simulation studies demonstrate the effectiveness of the proposed methods in identifying important variables, highlighting the role of the two-step method in reducing false positives and enhancing selection accuracy for Lasso, SCAD and MCP. Application to a real dataset further validates the practical utility of the proposed methods.