Extreme value copulas and max-stable processes

Mathieu Ribatet, Mohammed Sedki


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.

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SFdS / SMF - Journal de la Société Française de Statistique - ISSN 2102-6238