Kolmogorov and Wasserstein Distances between Max-Stable Distributions

Abstract

We derive explicit comparison bounds for multivariate max-stable distributions with unit-$\hat{I}\pm$-Fréchet margins. For the Kolmogorov distance, the bounds are expressed through Wasserstein distances between powered de Haan representers, total variation distances between angular measures, and discrepancies of the $\hat{I}¨$-functions in the inf--argmax decomposition. On the positive ${\ell }_{\hat{I}}\pm$-sphere, the coefficient multiplying the setwise angular total-variation distance contains no explicit dimension factor for the unnormalised angular measures used here. Separately, for $1\le p<\hat{I}\pm$, a synchronous de Haan--LePage coupling bounds the $p$-Wasserstein distance between the max-stable laws by an $\hat{I}\pm$-Wasserstein transport cost between their unpowered de Haan representers. We also compare laws with a common extreme-value copula and different Fréchet indices, obtaining an exact ${\ell }_1$-Wasserstein formula when $p=1$, and discuss applications to Archimax and clustered Archimax copulas and to Brown--Resnick/Hüsler--Reiss models.