Abstract : We are interested in the Wasserstein distance between two probability measures on $\R^n$ sharing the same copula $C$. The image of the probability measure $dC$ by the vectors of pseudo-inverses of marginal distributions is a natural generalization of the coupling known to be optimal in dimension $n=1$. It turns out that for cost functions $c(x,y)$ equal to the $p$-th power of the $L^q$ norm of $x-y$ in $\R^n$, this coupling is optimal only when $p=q$ i.e. when $c(x,y)$ may be decomposed as the sum of coordinate-wise costs.
https://hal-enpc.archives-ouvertes.fr/hal-00844906 Contributeur : Aurélien AlfonsiConnectez-vous pour contacter le contributeur Soumis le : mardi 16 juillet 2013 - 10:58:40 Dernière modification le : jeudi 20 janvier 2022 - 17:29:46 Archivage à long terme le : : jeudi 17 octobre 2013 - 04:15:38
Aurélien Alfonsi, Benjamin Jourdain. A remark on the optimal transport between two probability measures sharing the same copula. Statistics and Probability Letters, Elsevier, 2014, dx.doi.org/10.1016/j.spl.2013.09.035. ⟨hal-00844906⟩