Abstract: In this paper we propose a weighted composite quantile regression (WCQR) estimation approach and study model selection for nonlinear models with a diverging number of parameters. 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 oracle maximum likelihood estimator for a variety of error distributions including the normal, mixed-normal, Student's t, Cauchy distributions, etc. Based on the proposed WCQR, we use the adaptive-LASSO and SCAD regularization to simultaneously estimate parameters and select models. Under regularity conditions, we establish asymptotic equivalency of the two model selection methods and show that they perform as well as if the correct submodels are known in advance. We also suggest an algorithm for fast implementation of the proposed methodology. Simulations are conducted to compare different estimators, and an example is used to illustrate their performance.
Key words and phrases: Adaptive WCQR, adaptive LASSO, high dimensionality, model selection, oracle property, SCAD.