Estimation and variable selection for quantile partially linear single-index models

Abstract

Partially linear single-index models are flexible dimension reduction semiparametric tools yet still retain ease of interpretability as linear models. This paper is concerned with the estimation and variable selection for partially linear single-index quantile regression models. Polynomial splines are used to estimate the unknown link function. We first establish the asymptotic properties of the quantile regression estimators. For feature selection, we adopt the smoothly clipped absolute deviation penalty (SCAD) approach to select simultaneously single-index variables and partially linear variables. We show that the regularized variable selection estimators are consistent and possess oracle properties. The consistency and oracle properties are also established under the proposed linear approximation of the nonparametric link function that facilitates fast computation. Furthermore, we show that the proposed SCAD tuning parameter selectors via the Schwarz information criterion can consistently identify the true model. Monte Carlo studies and an application to Boston Housing price data are presented to illustrate the proposed approach.