L1-norm quantile regression

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 L-1-norm (LASSO) regularized quantile regression (L-1-norm QR), which uses the sum of the absolute values of the coefficients as the penalty. The L-1-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 L-1-norm QR. Furthermore, we derive an estimate for the effective dimension of the L-1-norm QR model, which allows convenient selection of the regularization parameter.