Distance Covariance, Independence, and Pairwise Differences
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866929410159935488 |
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| author | Raymaekers, Jakob Rousseeuw, Peter J. |
| author_facet | Raymaekers, Jakob Rousseeuw, Peter J. |
| contents | (To appear in The American Statistician.) Distance covariance (Székely, Rizzo, and Bakirov, 2007) is a fascinating recent notion, which is popular as a test for dependence of any type between random variables $X$ and $Y$. This approach deserves to be touched upon in modern courses on mathematical statistics. It makes use of distances of the type $|X-X'|$ and $|Y-Y'|$, where $(X',Y')$ is an independent copy of $(X,Y)$. This raises natural questions about independence of variables like $X-X'$ and $Y-Y'$, about the connection between Cov$(|X-X'|,|Y-Y'|)$ and the covariance between doubly centered distances, and about necessary and sufficient conditions for independence. We show some basic results and present a new and nontechnical counterexample to a common fallacy, which provides more insight. We also show some motivating examples involving bivariate distributions and contingency tables, which can be used as didactic material for introducing distance correlation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_13052 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distance Covariance, Independence, and Pairwise Differences Raymaekers, Jakob Rousseeuw, Peter J. Methodology (To appear in The American Statistician.) Distance covariance (Székely, Rizzo, and Bakirov, 2007) is a fascinating recent notion, which is popular as a test for dependence of any type between random variables $X$ and $Y$. This approach deserves to be touched upon in modern courses on mathematical statistics. It makes use of distances of the type $|X-X'|$ and $|Y-Y'|$, where $(X',Y')$ is an independent copy of $(X,Y)$. This raises natural questions about independence of variables like $X-X'$ and $Y-Y'$, about the connection between Cov$(|X-X'|,|Y-Y'|)$ and the covariance between doubly centered distances, and about necessary and sufficient conditions for independence. We show some basic results and present a new and nontechnical counterexample to a common fallacy, which provides more insight. We also show some motivating examples involving bivariate distributions and contingency tables, which can be used as didactic material for introducing distance correlation. |
| title | Distance Covariance, Independence, and Pairwise Differences |
| topic | Methodology |
| url | https://arxiv.org/abs/2406.13052 |