Robust variable selection of varying coefficient partially nonlinear model based on quantile regression

Quantile regression has been a popular topic for robust inference in semi-parametric models. However, there does not exist related literature for the varying coefficient partially nonlinear model (VCPNLM), which is the focus of this paper. Let alone on the quantile variable selection of VCPNLM. Specifically, via iteratively minimizing an average check loss estimation procedure based on quantile loss function, we propose a profile-type nonlinear quantile regression method for the VCPNLM, and further establish the asymptotic properties of the resulting estimators under some mild regularity conditions. In addition, to achieve sparsity when there exist irrelevant variables, we develop a variable selection procedure for high-dimensional VCPNLM by using the idea of shrinkage, and then demonstrate its oracle property. Two most important parameters including the smoothing parameter and the tuning parameter are also discussed, respectively. Finally, extensive numerical simulations with various errors are conducted to evaluate the finite sample performance of estimation and variable selection, and a real data analysis is further presented to illustrate the application of the proposed methods.