# Multivariate Rank-based Distribution-free Nonparametric Testing using Measure Transportation

arXiv: https://arxiv.org/abs/1909.08733 · HTML: https://mlsanity.com/paper/1909.08733/

## Key facts

- Cited by 24 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Sep 2019.

## Built on by (most cited first, 24 of 24)

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- [Deep Feature Screening: Feature Selection for Ultra High-Dimensional Data via Deep Neural Networks](https://mlsanity.com/paper/2204.01682/) (2022)
- [Multivariate rank via entropic optimal transport: sample efficiency and generative modeling](https://mlsanity.com/paper/2111.00043/) (2021)
- [Solving a Special Type of Optimal Transport Problem by a Modified Hungarian Algorithm](https://mlsanity.com/paper/2210.16645/) (2022)
- [A survey of some recent developments in measures of association](https://mlsanity.com/paper/2211.04702/) (2022)
- [On Rank Energy Statistics via Optimal Transport: Continuity, Convergence, and Change Point Detection](https://mlsanity.com/paper/2302.07964/) (2023)
- [Stability of Mean-Field Variational Inference](https://mlsanity.com/paper/2506.07856/) (2025)
- [A New Perspective On Denoising Based On Optimal Transport](https://mlsanity.com/paper/2312.08135/) (2023)
- [Fisher-Pitman permutation tests based on nonparametric Poisson mixtures with application to single cell genomics](https://mlsanity.com/paper/2106.03022/) (2021)
- [Perturbation-Assisted Sample Synthesis: A Novel Approach for Uncertainty Quantification](https://mlsanity.com/paper/2305.18671/) (2023)
- [A Copula Graphical Model for Multi-Attribute Data using Optimal Transport](https://mlsanity.com/paper/2404.06735/) (2024)
- [Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives](https://mlsanity.com/paper/2506.19025/) (2025)
- [Soft and subspace robust multivariate rank tests based on entropy regularized optimal transport](https://mlsanity.com/paper/2103.08811/) (2021)
- [Robust and efficient change point detection using novel multivariate rank-energy GoF test](https://mlsanity.com/paper/2111.00047/) (2021)
- [Bootstrapped Edge Count Tests for Nonparametric Two-Sample Inference Under Heterogeneity](https://mlsanity.com/paper/2304.13848/) (2023)
- [Multi-strategy Collaborative Optimized YOLOv5s and its Application in Distance Estimation](https://mlsanity.com/paper/2312.04113/) (2023)
- [On Ranking-based Tests of Independence](https://mlsanity.com/paper/2403.07464/) (2024)
- [Kernel Integrated R2: A Measure of Dependence](https://mlsanity.com/paper/2602.22985/) (2026)

## Abstract (arXiv)

In this paper, we propose a general framework for distribution-free nonparametric testing in multi-dimensions, based on a notion of multivariate ranks defined using the theory of measure transportation. Unlike other existing proposals in the literature, these multivariate ranks share a number of useful properties with the usual one-dimensional ranks; most importantly, these ranks are distribution-free. This crucial observation allows us to design nonparametric tests that are exactly distribution-free under the null hypothesis. We demonstrate the applicability of this approach by constructing exact distribution-free tests for two classical nonparametric problems: (i) testing for mutual independence between random vectors, and (ii) testing for the equality of multivariate distributions. In particular, we propose (multivariate) rank versions of distance covariance (Sz'ekely et al., 2007) and energy statistic (Sz'ekely and Rizzo, 2013) for testing scenarios (i) and (ii) respectively. In both these problems, we derive the asymptotic null distribution of the proposed test statistics. We further show that our tests are consistent against all fixed alternatives. Moreover, the proposed tests are tuning-free, computationally feasible and are well-defined under minimal assumptions on the underlying distributions (e.g., they do not need any moment assumptions). We also demonstrate the efficacy of these procedures via extensive simulations. In the process of analyzing the theoretical properties of our procedures, we end up proving some new results in the theory of measure transportation and in the limit theory of permutation statistics using Stein's method for exchangeable pairs, which may be of independent interest.