No-crossing single-index quantile regression curve estimation

Single-index quantile regression models can avoid the curse of dimensionality in nonparametric problems by assuming that the response is only related to a single linear combination of the covariates. Like the standard parametric or nonparametric quantile regression whose estimated curves may cross, the single-index quantile regression can also suffer quantile crossing, leading to an invalid distribution for the response. This issue has attracted considerable attention in the literature in recent years. In this paper, we consider single-index models, develop methods for quantile regression that guarantee non-crossing quantile curves, and extend the methods and results to composite quantile regression. The asymptotic properties of the proposed estimators are derived and their advantages over existing methods are explained. Simulation studies and a real data application are conducted to illustrate the finite sample performance of the proposed methods.