Nonparametric Multiple-Output Center-Outward Quantile Regression

Building on recent measure-transportation-based concepts of multivariate quantiles, we are considering the problem of nonparametric multiple-output quantile regression. Our approach defines nested conditional center-outward quantile regression contours and regions with given conditional probability content, the graphs of which constitute nested center-outward <i>quantile regression tubes</i> with given unconditional probability content; these (conditional and unconditional) probability contents do not depend on the underlying distribution—an essential property of quantile concepts. Empirical counterparts of these concepts are constructed, yielding interpretable empirical contours, regions, and tubes which are shown to consistently reconstruct (in the Pompeiu-Hausdorff topology) their population versions. Our method is entirely nonparametric and performs well in simulations—with possible heteroscedasticity and nonlinear trends. Its potential as a data-analytic tool is illustrated on some real datasets. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

基于近期提出的基于测度传输（measure-transportation）的多元分位数（multivariate quantiles）概念，本文聚焦非参数多输出分位数回归问题。本方法定义了带有给定条件概率含量的嵌套条件中心向外分位数回归等高线与区域，其图像可构成带有给定无条件概率含量的嵌套中心向外分位数回归管（quantile regression tubes）；此类（条件与无条件）概率含量不依赖于总体分布，这是分位数概念的核心属性。本文构建了上述概念的经验对应形式，得到可解释的经验等高线、区域与管，并证明其能够在庞皮奥-豪斯多夫拓扑（Pompeiu-Hausdorff topology）下一致地重构其总体形式。本方法完全属于非参数框架，在存在异方差（heteroscedasticity）与非线性趋势的仿真场景中仍表现出色。本文通过若干真实数据集展示了本方法作为数据分析工具的应用潜力。本文的补充材料可在线获取，其中包含可用于复现本研究的标准化材料说明。