Composite quantile regression and variable selection of the partial linear single-index models

In this paper, we propose a composite minimizing average check loss estimation(CMACLE)method for the composite quantile regression(CQR)of the partial linear single-index model(PLSIM)by local linear method. Based on constructive approach, the estimators by CMACLE are able to achieve the best convergence rate. The asymptotical normalities of the estimators are also derived. Meanwhile, the asymptotic efficiency of the CQR estimation relative to the mean regression are investigated. Further more, we propose a variable selection method for the CQR of PLSIM by combining the CMACLE procedure with the adaptive LASSO penalized method. The oracle properties of the proposed variable selection method are also established. Simulations with various non-normal errors and a real data analysis are conducted to assess the finite sample property of the proposed estimation and variable selection methods.