Regression Estimator for the Tail Index

Abstract

Estimating the tail index parameter is one of the primal objectives in extreme value theory. For heavy-tailed distributions the Hill estimator is the most popular way to estimate this parameter. Several recent publications’ aim was to improve the Hill estimator, using different methods, for example the bootstrap, or the Kolmogorov–Smirnov metric. These methods are asymptotically consistent, but for tail index ξ> 0.5 the estimations fail to approach the theoretical value for realistic sample sizes. In this paper, we introduce new empirical methods, which combine the advantages of the Kolmogorov–Smirnov approach and the bootstrap. We demonstrate that our estimators are able to estimate large tail index parameters well and might also be useful for relatively small sample sizes. As an application, we consider the classic Danish fire data set and the most destructive natural disasters in Europe.