Garrotized kernel machine in semiparametric quantile regression

Quantile regression has been attractive due to its interpretability and robustness, which is a useful tool in regression analysis. Moreover, the relationship between response and some covariates is complex nonlinearity in real data, and there probably exists high-order interaction effects between covariates. Furthermore, the irrelevant covariates included in the model lead to unsatisfactory prediction results. Inspired by this, we propose a novel estimation and variable selection method of the semiparametric quantile regression. The unknown function is estimated by the kernel machine technique, and the LASSO penalty is applied to achieve variable selection in the nonlinear part. Importantly, we introduce slack variables to solve the limitations of the quantile loss function in optimization problems, and solve the optimization problem of nonlinear functions through local linear approximation technology. The proposed estimator is easy-to-implement via an efficient cyclical coordinate descent algorithm. Both simulations and real applications demonstrate the convincing performance of the proposed estimator.