Weighted composite quantile regression for single-index models

In this paper we propose a weighted composite quantile regression (WCQR) estimation for single-index models. For parametric part, the WCQR is augmented using a data-driven weighting scheme. With the error distribution unspecified, the proposed estimators share robustness from quantile regression and achieve nearly the same efficiency as the semiparametric maximum likelihood estimator for a variety of error distributions including the Normal, Student's t, Cauchy distributions, etc. Furthermore, based on the proposed WCQR, we use the adaptive-LASSO to study variable selection for parametric part in the single-index models. For nonparametric part, the WCQR is augmented combining the equal weighted estimators with possibly different weights. Because of the use of weights, the estimation bias is eliminated asymptotically. By comparing asymptotic relative efficiency theoretically and numerically, WCQR estimation all outperforms the CQR estimation and some other estimate methods. Under regularity conditions, the asymptotic properties of the proposed estimations are established. The simulation studies and two real data applications are conducted to illustrate the finite sample performance of the proposed methods.