Kolmogorov and Wasserstein Distances between Max-Stable Distributions

Abstract

We derive explicit comparison bounds for multivariate max-stable distributions with unit-$\alpha$-Fr\'echet 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 $\Psi$-functions in the inf--argmax decomposition. On the positive $\ell_\alpha$-sphere, the coefficient multiplying the setwise angular total-variation distance contains no explicit dimension factor for the unnormalised angular measures used here. Separately, for $1\le p<\alpha$, a synchronous de Haan--LePage coupling bounds the $p$-Wasserstein distance between the max-stable laws by an $\alpha$-Wasserstein transport cost between their unpowered de Haan representers. We also compare laws with a common extreme-value copula and different Fr\'echet indices, obtaining an exact $\ell_1$-Wasserstein formula when $p=1$, and discuss applications to Archimax and clustered Archimax copulas and to Brown--Resnick/H\"usler--Reiss models.