On Asymptotic Properties of the Separating Hill Estimator

TL;DR

This paper analyzes the asymptotic properties of the separating Hill estimator in multivariate extreme value analysis, demonstrating its consistency and normality when location and scatter are estimated, extending prior simulation-based findings.

Contribution

It provides the first analytical proof of the asymptotic behavior of the separating Hill estimator with estimated parameters, under mild conditions.

Findings

The estimator is consistent with estimated location and scatter.

It is asymptotically normal under mild conditions.

Analytical results extend previous simulation studies.

Abstract

Modeling and understanding multivariate extreme events is challenging, but of great importance in various applications - e.g. in biostatistics, climatology, and finance. The separating Hill estimator can be used in estimating the extreme value index of a heavy tailed multivariate elliptical distribution. We consider the asymptotic behavior of the separating Hill estimator under estimated location and scatter. The asymptotic properties of the separating Hill estimator are known under elliptical distribution with known location and scatter. However, the effect of estimation of the location and scatter has previously been examined only in a simulation study. We show, analytically, that the separating Hill estimator is consistent and asymptotically normal under estimated location and scatter, when certain mild conditions are met.