# Extremal attractors of Liouville copulas

## TL;DR

This paper derives new extremal models called scaled extremal Dirichlet from Liouville copulas, providing tractable formulas and algorithms for simulation and inference, and applies them to river flow data.

## Contribution

It introduces the scaled extremal Dirichlet models as new max-stable limits of Liouville copulas, with explicit formulas and efficient algorithms for analysis.

## Key findings

- Derived the limiting extremal copulas for Liouville copulas.
- Provided tractable formulas for dependence structures and inference.
- Applied models successfully to river flow data.

## Abstract

Liouville copulas, which were introduced in McNeil and Neslehova (2010), are asymmetric generalizations of the ubiquitous Archimedean copula class. They are the dependence structures of scale mixtures of Dirichlet distributions, also called Liouville distributions. In this paper, the limiting extreme-value copulas of Liouville copulas and of their survival counterparts are derived. The limiting max-stable models, termed here the scaled extremal Dirichlet, are new and encompass several existing classes of multivariate max-stable distributions, including the logistic, negative logistic and extremal Dirichlet. As shown herein, the stable tail dependence function and angular density of the scaled extremal Dirichlet model have a tractable form, which in turn leads to a simple de Haan representation. The latter is used to design efficient algorithms for unconditional simulation based on the work of Dombry, Engelke and Oesting (2015) and to derive tractable formulas for maximum-likelihood inference. The scaled extremal Dirichlet model is illustrated on river flow data of the river Isar in southern Germany.