Multivariate nonparametric estimation of the Pickands dependence function using Bernstein polynomials

Abstract

Many applications in risk analysis require the estimation of the dependence among multivariate maxima, especially in environmental sciences. Such dependence can be described by the Pickands dependence function of the underlying extreme-value copula. Here, a nonparametric estimator is constructed as the sample equivalent of a multivariate extension of the madogram. Shape constraints on the family of Pickands dependence functions are taken into account by means of a representation in terms of Bernstein polynomials. The large-sample theory of the estimator is developed and its finite-sample performance is evaluated with a simulation study. The approach is illustrated with a dataset of weekly maxima of hourly rainfall in France recorded from 1993 to 2011 at various weather stations all over the country. The stations are grouped into clusters of seven stations, where our interest is in the extremal dependence within each cluster.