Practical Notes On Multivariate Modeling Based on Elliptical Copulas

Wang, Xiaojing; Yan, Jun

Journal de la société française de statistique, Numéro spécial sur les copules, Tome 154 (2013) no. 1, pp. 102-115.

Les distributions multivariées construites à partir de copules elliptiques sont utilisées dans de nombreux domaines comme l’hydrologie et la finance. Nous nous intéressons à deux aspects pratiques concernant ces modèles. Dans un premier temps, nous attirons l’attention sur l’importance de la propriété de consistance définie par Kano (1994, Journal of Multivariate Analysis, 51 :139–147). Certaines distributions elliptiques ne satisfont pas cette propriété, ce qui limite les applications et l’implantation des copules correspondantes dans les logiciels. Dans un deuxième temps, nous donnons deux méthodes conditionnelles pour la génération d’échantillons aléatoires à partir de distributions elliptiques. La première approche présentée est fondée sur une représentation stochastique des distributions elliptiques alors que la seconde utilise une méthode d’acceptation/rejet. L’utilisation des deux méthodes est illustrée dans la cadre de la modélisation de données hydrologiques trivariées.

Multivariate distributions based on elliptical copulas have been widely used in many fields such as hydrology and finance. We focus on two practical issues of applications of such models. The first is a caveat rooted in a consistency property defined by Kano (1994, Journal of Multivariate Analysis, 51:139–147) for elliptical distributions. Some elliptical families do not have this property, which puts practical limitations on applications and software implementation of the corresponding elliptical copulas. The second issue is on conditional sampling from such distributions, which is important in Monte Carlo statistical inferences, especially when closed-form solutions are not available or feasible. Two sampling methods are presented: a direct sampling approach based on a stochastic representation of elliptical distributions, and an acceptance/rejection sampling method. The latter also provides an importance sampler as a byproduct, which may have higher efficiency for some applications. A trivariate model of the volume, duration, and peak intensity of annual extreme storms illustrates the sampling algorithms.

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