L₁-Norm Quantile Regression

Abstract

Classical regression methods have focused mainly on estimating conditional mean functions. In recent years, however, quantile regression has emerged as a comprehensive approach to the statistical analysis of response models. In this article we consider the L₁-norm (LASSO) regularized quantile regression (L₁-norm QR), which uses the sum of the absolute values of the coefficients as the penalty. The L₁-norm penalty has the advantage of simultaneously controlling the variance of the fitted coefficients and performing automatic variable selection. We propose an efficient algorithm that computes the entire solution path of the L₁-norm QR. Furthermore, we derive an estimate for the effective dimension of the L₁-norm QR model, which allows convenient selection of the regularization parameter.

Key Words:

• Effective dimension
• LASSO
• Linear programming
• L₁-norm penalty
• Variable selection