On the Kolmogorov Distance of Max-Stable Distributions

TL;DR

This paper establishes explicit bounds on the Kolmogorov distance for multivariate max-stable distributions with Fréchet margins, using Wasserstein, total variation, and Psi-function discrepancies, with applications to various models.

Contribution

It provides new explicit bounds on the Kolmogorov distance for max-stable distributions, improving previous results by removing the dimension factor and extending to different margins.

Findings

Derived bounds in terms of Wasserstein, total variation, and Psi-function discrepancies.

Applicable to logistic, comonotonic, independent, and Brown-Resnick models.

Extensions to different margins and copulas discussed.

Abstract

In this contribution, we derive explicit bounds on the Kolmogorov distance for multivariate max-stable distributions with Fr\'echet margins. We formulate those bounds in terms of (i) Wasserstein distances between de Haan representers, (ii) total variation distances between spectral/angular measures - removing the dimension factor from earlier results in the canonical sphere case - and (iii) discrepancies of the Psi-functions in the inf-argmax decomposition. Extensions to different margins and Archimax/clustered Archimax copulas are further discussed. Examples include logistic, comonotonic, independent and Brown-Resnick models.