Multivariate Extreme Value Index: Hill and Moment-Based Estimators

Modeling extreme events is of paramount importance in various areas of science biostatistics, climatology, finance, geology, and telecommunications, to name a few. Most of these application areas involve multivariate data. Estimation of the extreme value index plays a crucial role in modeling rare events. We propose two extensions of the Hill estimator for the tail index of a regular varying elliptical random vector. The first one is based on the distance between a tail probability contour and the observations outside this contour. We denote this class as separating estimators. The second one is based on the norm of an arbitrary order and denoted as the class of angular estimators. However, the Hill estimator is only suitable for heavy tailed distributions. As in the case of the separating multivariate Hill estimator, we consider estimation of the extreme value index under the assumption of multivariate ellipticity. We provide affine invariant multivariate generalizations of the moment estimator and the mixed moment estimator. These estimators are suitable for both: light and heavy tailed distributions. Asymptotic properties of the new extreme value index estimators are derived under multivariate elliptical distribution with known location and scatter. The effect of replacing true location and scatter by estimates is examined in a thorough simulation study.

This talk is based on two papers:

Dominicy Y., Ilmonen P. and Veredas D. (2016). Multivariate Hill Estimators, forthcoming in the International Statistical Review.

Heikkila M., Dominicy Y. and Ilmonen Pauliina (2016). Multivariate Moment Based Extreme Value Index Estimators, submitted.