Multivariate Distribution-Free Nonparametric Testing: Generalizing Wilcoxon's Tests via Optimal Transport

This paper reviews recent advancements in the application of optimal transport (OT) to multivariate distribution-free nonparametric testing. Inspired by classical rank-based methods, such as Wilcoxon's rank-sum and signed-rank tests, we explore how OT-based ranks and signs generalize these concepts to multivariate settings, while preserving key properties, including distribution-freeness, robustness, and efficiency. Using the framework of asymptotic relative efficiency (ARE), we compare the power of the proposed (generalized Wilcoxon) tests against the Hotelling's $T^2$ test. The ARE lower bounds reveal the Hodges-Lehmann and Chernoff-Savage phenomena in the context of multivariate location testing, underscoring the high power and efficiency of the proposed methods. We also demonstrate how OT-based ranks and signs can be seamlessly integrated with more modern techniques, such as kernel methods, to develop universally consistent, distribution-free tests. Additionally, we present novel results on the construction of consistent and distribution-free kernel-based tests for multivariate symmetry, leveraging OT-based ranks and signs.