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

Betina Berghaus, Axel Bücher, Holger Dette


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