Résumé : Modeling extreme events is of paramount importance in various areas ofscience — biostatistics, climatology, finance, geology, and telecommunications, toname a few. Most of these application areas involve multivariate data. Estimationof the extreme value index plays a crucial role in modeling rare events. There isan affine invariant multivariate generalization of the well known Hill estimator—theseparating Hill estimator. However, the Hill estimator is only suitable for heavy taileddistributions. As in the case of the separating multivariate Hill estimator, we considerestimation of the extreme value index under the assumption of multivariate ellipticity.We provide affine invariant multivariate generalizations of the moment estimator andthe mixed moment estimator. These estimators are suitable for both: light and heavytailed distributions. Asymptotic properties of the new extreme value index estimatorsare derived under multivariate elliptical distribution with known location and scatter.The effect of replacing true location and scatter by estimates is examined in a thoroughsimulation study.