Extreme value copulas and max-stable processes

Abstract

During the last decades, copulas have been increasingly used to model the dependence across several random variables such as the joint modelling of the intensity and the duration of rainfall storms. When the problem consists in modelling extreme values, i.e., only the tails of the distribution, the extreme value theory tells us that one should consider max-stable distributions and put some restrictions on the copulas to be used. Although the theory for multivariate extremes is well established, its foundation is usually introduced outside the copula framework. This paper tries to unify these two frameworks in a single view. Moreover the latest developments on spatial extremes and max-stable processes will be introduced. At first glance the use of copulas for spatial problems sounds a bit odd but since usually stochastic processes are observed at a finite number of locations, the inferential procedure is intrinsically multivariate. An application on the spatial modelling of extreme temperatures in Switzerland is given. Results show that the use of non extreme value based models can largely underestimate the spatial dependence and the assumptions made on the spatial dependence structure should be chosen with care.