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.

单指标分位数回归模型（single-index quantile regression）通过假定响应变量仅与协变量的单一线性组合相关，可规避非参数问题中的维数灾难。与标准参数化或非参数化分位数回归类似，其估计曲线可能出现交叉，单指标分位数回归同样会遭遇分位数交叉问题，进而导致响应变量的分布不具备有效性。近年来，该问题已受到学术界的广泛关注。本文针对单指标模型展开研究，提出可确保分位数曲线无交叉的分位数回归方法，并将该方法与相关结论推广至复合分位数回归（composite quantile regression）。我们推导了所提出估计量的渐近性质，并阐释了其相较于现有方法的优势。通过模拟实验与真实数据集应用，验证了所提方法的有限样本表现。