Multivariate Distribution-Free Nonparametric Testing: Generalizing Wilcoxon's Tests via Optimal Transport
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866929761620590592 |
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| author | Huang, Zhen Sen, Bodhisattva |
| author_facet | Huang, Zhen Sen, Bodhisattva |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12236 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multivariate Distribution-Free Nonparametric Testing: Generalizing Wilcoxon's Tests via Optimal Transport Huang, Zhen Sen, Bodhisattva Methodology 62G10, 62G20, 62H15 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. |
| title | Multivariate Distribution-Free Nonparametric Testing: Generalizing Wilcoxon's Tests via Optimal Transport |
| topic | Methodology 62G10, 62G20, 62H15 |
| url | https://arxiv.org/abs/2503.12236 |