Minimum distance estimators of the Pickands dependence function and related tests of multivariate extreme-value dependence

Berghaus, Betina; Bücher, Axel; Dette, Holger

Journal de la société française de statistique, Numéro spécial sur les copules, Tome 154 (2013) no. 1, pp. 116-137.

Nous nous intéressons à l’estimation de la fonction de dépendance de Pickands correspondant à une distribution de valeurs extrêmes multivariée. Un estimateur du minimum de distance fondé sur la distance $L^2$ entre les logarithmes de la copule empirique et de la copule inconnue est proposé et sa convergence faible est démontrée. Contrairement à d’autres procédures récemment proposées dans la littérature pour l’estimation de la fonction de dépendance de Pickands multivariée (voir [ Zhang et al., 2008 ] et [ Gudendorf and Segers, 2011 ]), les estimateurs étudiés dans ce travail ne requièrent pas la donnée des distributions marginales et sont ainsi une alternative à la méthode de [ Gudendorf and Segers, 2012 ]. De plus, l’approche du minimum de distance considérée permet naturellement la construction d’un test d’appartenance à la classe des copules de valeurs extrêmes dont la consistance est démontrée pour les copules modélisant une association positive. Des simulations sont enfin utilisées pour étudier empiriquement, sur des échantillons de taille finie, les propriétés de l’estimateur du minimum de distance ainsi que du test associé mis en oeuvre à l’aide d’un rééchantillonnage fondé sur des multiplicateurs.

We consider the problem of estimating the Pickands dependence function corresponding to a multivariate extreme-value distribution. A minimum distance estimator is proposed which is based on an $L^2$-distance between the logarithms of the empirical and the unknown extreme-value copula. The minimizer can be expressed explicitly as a linear functional of the logarithm of the empirical copula and weak convergence of the corresponding process on the simplex is proved. In contrast to other procedures which have recently been proposed in the literature for the nonparametric estimation of a multivariate Pickands dependence function (see [ Zhang et al., 2008 ] and [ Gudendorf and Segers, 2011 ]), the estimators constructed in this paper do not require knowledge of the marginal distributions and are an alternative to the method which has recently been suggested in [ Gudendorf and Segers, 2012 ]. Moreover, the minimum distance approach allows the construction of a simple test for the hypothesis of a multivariate extreme-value copula, which is consistent against a broad class of alternatives. The finite-sample properties of the estimator and a multiplier bootstrap version of the test are investigated by means of a simulation study.

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