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.
@article{1013203459,
author = {Einmahl, John H.J. and de Haan, Laurens and Piterbarg, Vladimir I.},
title = {Nonparametric estimation of the spectral measure of an extreme
value distribution},
journal = {Ann. Statist.},
volume = {29},
number = {2},
year = {2001},
pages = { 1401-1423},
language = {en},
url = {http://dml.mathdoc.fr/item/1013203459}
}
Einmahl, John H.J.; de Haan, Laurens; Piterbarg, Vladimir I. Nonparametric estimation of the spectral measure of an extreme
value distribution. Ann. Statist., Tome 29 (2001) no. 2, pp. 1401-1423. http://gdmltest.u-ga.fr/item/1013203459/