Why the 1-Wasserstein distance is the area between the two marginal CDFs

We elucidate why the 1-Wasserstein distance \(W_1\) coincides with the area between the two marginal cumulative distribution functions (CDFs). We first describe the Wasserstein distance in terms of copulas, and then show that \(W_1\) with the Euclidean distance is attained with the \(M\) copula. Two...

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Bibliographic Details
Published inarXiv.org
Main Authors De Angelis, Marco, Gray, Ander
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 05.11.2021
Subjects
Distribution functions
Euclidean geometry
Random variables
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Summary:We elucidate why the 1-Wasserstein distance \(W_1\) coincides with the area between the two marginal cumulative distribution functions (CDFs). We first describe the Wasserstein distance in terms of copulas, and then show that \(W_1\) with the Euclidean distance is attained with the \(M\) copula. Two random variables whose dependence is given by the \(M\) copula manifest perfect (positive) dependence. If we express the random variables in terms of their CDFs, it is intuitive to see that the distance between two such random variables coincides with the area between the two CDFs.
ISSN:2331-8422