No-Crossing Single-Index Quantile Regression Curve Estimation

Single-index quantile regression (QR) 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 QR whose estimated curves may cross, the single-index QR can also suffer quantile crossing, leading to an invalid distribution for the response. This issue has attracted considerable attention in the literature in the recent year. In this article, we consider single-index models, develop methods for QR that guarantee noncrossing 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（Quantile Regression, QR）模型通过假设响应变量仅与协变量的单一线性组合相关，得以规避非参数问题中的维数灾难。与估计曲线可能出现交叉的标准参数化或非参数化分位数回归类似，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失去有效性。近年来，该问题已受到学界的广泛关注。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）领域。本文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，本文验证了所提方法的有限样本表现。