Extreme value copulas and max-stable processes

Ribatet, Mathieu; Sedki, Mohammed

Journal de la société française de statistique, Numéro spécial sur les copules, Tome 154 (2013) no. 1, pp. 138-150.

Les dernières décennies ont vu une utilisation des copules de plus en plus fréquente afin de modéliser la dépendance présente au sein d’un groupe de plusieurs variables aléatoires ; par exemple afin de modéliser simultanément l’intensité et la durée d’un événement pluvieux. Lorsque l’intérêt porte sur la modélisation des valeurs extrêmes, i.e., seulement les queues de la distribution, la théorie des valeurs extrêmes nous dicte quelles distributions considérer. Ces dernières doivent être max-stables et imposent donc des contraintes sur les copules adéquates. Bien que la théorie pour les extrêmes multivariées soit bien établie, elle est généralement introduite en dehors du cadre des copules. Ce papier essaye de présenter la théorie des valeurs extrêmes par le monde des copules. Les derniers développements sur les extrêmes spatiaux et les processus max-stables seront également évoqués. Bien qu’il paraisse étrange au premier abord de parler de copules pour les processus stochastiques, leur utilisation peut être adéquate puisque les processus sont souvent observés en un nombre fini de positions et la procédure d’estimation est alors intrinsèquement multivariée. Une application à la modélisation spatiale des températures extrêmes en Suisse est donnée. Les résultats montrent que l’utilisation de modèles non extrêmes peut largement sous-estimer la dépendance spatiale et que le choix fait sur la structure de dépendance spatiale est primordial.

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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