Robust check loss-based inference of semiparametric models and its application in environmental data

Abstract

Composite quantile regression has been shown to be a more competitive method than the well-known mean regression and quantile regression approaches, since it can significantly improve the estimation efficiency for a wide class of errors. In this paper, we propose a profiling composite quantile regression for robust estimation of semiparametric varying-coefficient partially linear model (VCPLM) by combining the ideas of local linear composite quantile regression with the profile technique. Different from the step-wise estimation method considered in Kai et al. (2011), an iterative algorithm is developed for the estimates of parametric and nonparametric functions. Under some mild regularity conditions, we establish the asymptotic normalities of the obtained estimators and further briefly discuss the related asymptotic relative efficiency. Noteworthy, an entirely different way from Kai et al. (2011) is presented to establish the asymptotic normality of the parametric component, which is frequently of the primary interest in VCPLM. Some Monte Carlo simulations with various errors and an environmental data application are conducted to evaluate the finite sample performance of the proposed methodology.