Nonparametric estimation of the spectral measure of an extreme value distribution

Abstract

Let $(\mathcal{X}_1, \mathcal{Y}_1),\dots,(\mathcal{X}_n, \mathcal{Y}_n)$ be a random sample from a bivariate distribution function $F$ in the domain of max-attraction of a distribution function $G$. This $G$ is characterised by the two extreme value indices and its spectral or angular measure. The extreme value indices determine both the marginals and the spectral measure determines the dependence structure of $G$. One of the main issues in multivariate extreme value theory is the estimation of this spectral measure. We construct a truly nonparametric estimator of the spectral measure, based on the ranks of the above data. Under natural conditions we prove consistency and asymptotic normality for the estimator. In particular,the result is valid for all values of the extreme value indices. The theory of (local) empirical processes is indispensable here. The results are illustrated by an application to real data and a small simulation study.