Three Applications of Measure Transportation in Statistical Inference

Measure transportation recently helped solve some long-standing open problems in statistical inference by providing satisfactory definitions of ranks and quantiles beyond the classical univariate case. We describe three examples of this fact: Wilcoxon-type distribution-free rank-based tests for the multivariate two-sample problem with unspecified error densities, nonparametric multiple-output quantile regression, and Cramér-von Mises Goodness-of-Fit tests for directional data.