Wasserstein distance in terms of the comonotonicity Copula
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910670496202752 |
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| author | Abdellatif, Mariem Kuching, Peter Rüdiger, Barbara Ventura, Irene |
| author_facet | Abdellatif, Mariem Kuching, Peter Rüdiger, Barbara Ventura, Irene |
| contents | The aim of this article is to write the $p$-Wasserstein metric $W_p$ with the $p$-norm, $p\in [1,\infty)$, on $\R^d$ in terms of copula. In particular for the case of one-dimensional distributions, we get that the copula employed to get the optimal coupling of the Wasserstein distances is the comotonicity copula. We obtain the equivalent result also for $d$-dimensional distributions under the sufficient and necessary condition that these have the same dependence structure of their one-dimensional marginals, i.e that the $d$-dimensional distributions share the same copula. Assuming $p\neq q$, $p,q$ $\in [1,\infty)$ and that the probability measures $μ$ and $ν$ are sharing the same copula, we also analyze the Wasserstein distance $W_{p,q}$ discussed in \cite{Alfonsi} and get an upper and lower bounds of $W_{p,q}$ in terms of $W_p$, written in terms of comonotonicity copula. We show that as a consequence the lower and upper bound of $W_{p,q}$ can be written in terms of generalized inverse functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19914 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wasserstein distance in terms of the comonotonicity Copula Abdellatif, Mariem Kuching, Peter Rüdiger, Barbara Ventura, Irene Statistics Theory Probability The aim of this article is to write the $p$-Wasserstein metric $W_p$ with the $p$-norm, $p\in [1,\infty)$, on $\R^d$ in terms of copula. In particular for the case of one-dimensional distributions, we get that the copula employed to get the optimal coupling of the Wasserstein distances is the comotonicity copula. We obtain the equivalent result also for $d$-dimensional distributions under the sufficient and necessary condition that these have the same dependence structure of their one-dimensional marginals, i.e that the $d$-dimensional distributions share the same copula. Assuming $p\neq q$, $p,q$ $\in [1,\infty)$ and that the probability measures $μ$ and $ν$ are sharing the same copula, we also analyze the Wasserstein distance $W_{p,q}$ discussed in \cite{Alfonsi} and get an upper and lower bounds of $W_{p,q}$ in terms of $W_p$, written in terms of comonotonicity copula. We show that as a consequence the lower and upper bound of $W_{p,q}$ can be written in terms of generalized inverse functions. |
| title | Wasserstein distance in terms of the comonotonicity Copula |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2410.19914 |