Nonparametric Multiple-Output Center-Outward Quantile Regression

Abstract–

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 quantile regression tubes 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 non-parametric and performs well in simulations—with possible heteroskedasticity and nonlinear trends. Its potential as a data-analytic tool is illustrated on some real datasets.

Keywords:

• Multiple-output regression
• Center-outward quantiles
• Optimal transport.

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