Markov Kernels and the Conditional Extreme Value Model

TL;DR

This paper explores a flexible multivariate extreme value model based on conditional distributions, extending classical methods by using convergence of measures and multivariate regular variation to better approximate risk regions.

Contribution

It places the conditional extreme value model within a broader framework using measure convergence and regular variation, addressing previous technical limitations.

Findings

The model allows for components not in the domain of attraction.

It provides an approximation for probabilities of risk regions.

The approach generalizes classical multivariate extreme value theory.

Abstract

The classical approach to multivariate extreme value modelling assumes that the joint distribution belongs to a multivariate domain of attraction. This requires each marginal distribution be individually attracted to a univariate extreme value distribution. An apparently more flexible extremal model for multivariate data was proposed by Heffernan and Tawn under which not all the components are required to belong to an extremal domain of attraction but assumes instead the existence of an asymptotic approximation to the conditional distribution of the random vector given one of the components is extreme. Combined with the knowledge that the conditioning component belongs to a univariate domain of attraction, this leads to an approximation of the probability of certain risk regions. The original focus on conditional distributions had technical drawbacks but is natural in several contexts. We place this approach in the context of the more general approach using convergence of measures and multivariate regular variation on cones.