Simple models for multivariate regular variations and the H\"usler-Reiss Pareto distribution

TL;DR

This paper simplifies the modeling of multivariate extreme values by revisiting regular variations, recovering key distributions, and analyzing the H"usler-Reiss Pareto model's exponential family structure and estimation methods.

Contribution

It introduces a simple framework for multivariate extreme value distributions and explores the exponential family properties of the H"usler-Reiss Pareto model, including estimation and hypothesis testing.

Findings

Recovered major multivariate extreme value distributions using a simple framework

Identified the exponential family structure of the H"usler-Reiss Pareto model

Developed maximum likelihood estimation and likelihood ratio tests for tail index variations

Abstract

We revisit multivariate extreme value theory modeling by emphasizing multivariate regular variations and the multivariate Breiman Lemma. This allows us to recover in a simple framework the most popular multivariate extreme value distributions, such as the logistic, negative logistic, Dirichlet, extremal-$t$ and H\"usler-Reiss models. In a second part of the paper, we focus on the H\"usler-Reiss Pareto model and its surprising exponential family property. After a thorough study of this exponential family structure, we focus on maximum likelihood estimation. We also consider the generalized H\"usler-Reiss Pareto model with different tail indices and a likelihood ratio test for discriminating constant tail index versus varying tail indices.