Multivariate extremes and dependence structures : a theoretical background for modelling

Abstract

Many fields of modern science have to deal with events which are rare but of outstanding importance. Extreme value theory is a practical and useful mathematical tool for modelling events which occur with very small probability. In a wide variety of applications these extreme events have an inherently multivariate character. This thesis provides an overview of the relevant theoretical results for modelling multivariate extremes and their dependence structures. We study multivariate extreme value distributions (MEVDs) and characterise their maximum domain of attraction (MDA). We state the relationships between four equivalent representations of MEVDs which can be used as a basis for estimation. Moreover we look at tail dependence coefficients and provide information about the underlying dependence. The central result is the multivariate extension of the Fisher-Tippett Theorem which basically says that the maximum domain of attraction of a MEVD is characterised by the univariate MDAs of its margins and a so-called copula domain of attraction (CDA) of its copula. We construct explicit examples of copulas which are in no CDA and describe models for the tail of a multivariate distribution function. In order to facilitate model building, some methods to construct new extreme value copulas from known ones are presented.