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

De Angelis, Marco and Gray, Ander (2022) Why the 1-Wasserstein distance is the area between the two marginal CDFs. Preprint / Working Paper. arXiv, Ithaca, New York. (https://doi.org/10.48550/arXiv.2111.03570)

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Abstract

We elucidate why the 1-Wasserstein distance W1 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 W1 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.

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