Improved estimation of the extreme value index using related variables

TL;DR

This paper introduces an improved estimator for the extreme value index that leverages related variables and tail dependence, demonstrating enhanced accuracy through theoretical analysis, simulations, and an earthquake loss case study.

Contribution

It proposes an adapted Hill estimator utilizing related variables and tail dependence, significantly improving tail heaviness estimation in heavy-tailed data.

Findings

The adapted estimator shows superior performance over the classical Hill estimator.

Asymptotic normality of the new estimator is established.

Simulation results confirm improved estimation accuracy.

Abstract

Heavy tailed phenomena are naturally analyzed by extreme value statistics. A crucial step in such an analysis is the estimation of the extreme value index, which describes the tail heaviness of the underlying probability distribution. We consider the situation where we have next to the $n$ observations of interest another $n+m$ observations of one or more related variables, like, e.g., financial losses due to earthquakes and the related amounts of energy released, for a longer period than that of the losses. Based on such a data set, we present an adapted version of the Hill estimator that shows greatly improved behavior and we establish the asymptotic normality of this estimator. For this adaptation the tail dependence between the variable of interest and the related variable(s) plays an important role. A simulation study confirms the substantially improved performance of our adapted estimator relative to the Hill estimator. We also present an application to the aforementioned earthquake losses.