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.