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On the Kolmogorov Distance of Max-Stable Distributions

Enkelejd Hashorva

TL;DR

The work tackles the problem of quantifying the difference between two $d$-dimensional max-stable distributions with unit-$α$-Fréchet margins by bounding the Kolmogorov distance $d_K(F_1,F_2)$. It develops explicit, computable bounds in terms of Wasserstein distances between de Haan representers, total variation between angular measures, and discrepancies of the $Ψ$-functions in the inf-argmax decomposition, with extensions to nonstandard margins and Archimax/clustered Archimax copulas. The results include dimension-free TV bounds, refined $W_2$-based bounds for log-exponential (Brown–Resnick) models, and explicit bounds for a variety of models such as comonotonic, independent, and logistic families, as well as for Archimax constructions. Overall, the paper provides practical, model-agnostic tools for comparing multivariate EVT models and for uncertainty quantification in extreme-value analysis, with potential applications in statistical inference and risk assessment.

Abstract

In this contribution, we derive explicit bounds on the Kolmogorov distance for multivariate max-stable distributions with Fréchet 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.

On the Kolmogorov Distance of Max-Stable Distributions

TL;DR

The work tackles the problem of quantifying the difference between two -dimensional max-stable distributions with unit--Fréchet margins by bounding the Kolmogorov distance . It develops explicit, computable bounds in terms of Wasserstein distances between de Haan representers, total variation between angular measures, and discrepancies of the -functions in the inf-argmax decomposition, with extensions to nonstandard margins and Archimax/clustered Archimax copulas. The results include dimension-free TV bounds, refined -based bounds for log-exponential (Brown–Resnick) models, and explicit bounds for a variety of models such as comonotonic, independent, and logistic families, as well as for Archimax constructions. Overall, the paper provides practical, model-agnostic tools for comparing multivariate EVT models and for uncertainty quantification in extreme-value analysis, with potential applications in statistical inference and risk assessment.

Abstract

In this contribution, we derive explicit bounds on the Kolmogorov distance for multivariate max-stable distributions with Fréchet 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.
Paper Structure (11 sections, 6 theorems, 135 equations)

This paper contains 11 sections, 6 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation.
  4. Main Results
  5. Discussions and Examples
  6. Non-standard margins
  7. Different margins
  8. Kolmogorov distance of Archimax and clustered Archimax copulas
  9. General 1-homogeneous $\tau$ maps
  10. Examples
  11. Proofs

Key Result

Lemma 1

Let $(\mathsf X,\rho)$ be a compact metric space with diameter $D:=\sup_{x,y\in\mathsf X}\rho(x,y)<\infty$. Then for Borel probability measures $\mu,\nu$ on $\mathsf X$ we have

Theorems & Definitions (16)

  • Lemma 1: Kantorovich–Rubinstein and TV comparison
  • Remark 1
  • Lemma 2
  • Remark 2
  • Theorem 3.1
  • Remark 3
  • Corollary 1
  • Remark 4
  • Theorem 3.2
  • Corollary 2
  • ...and 6 more