Regression estimator for the tail index

TL;DR

This paper introduces a new empirical method for estimating the tail index in extreme value theory, especially effective for high tail indices and small samples, improving upon existing estimators like Hill.

Contribution

The paper proposes a novel empirical estimator for the tail index that performs well for high values and small sample sizes, addressing limitations of existing methods.

Findings

New estimator accurately estimates high tail indices

Performs better with small sample sizes

Outperforms some existing methods in simulations

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 the tail index parameter. Improving the Hill estimator was aimed by recent works with different methods, for example by using bootstrap, or Kolmogorov-Smirnov metric. These methods are asymptotically consistent, but for tail index $\xi >1$ and smaller sample sizes the estimation fails to approach the theoretical value for realistic sample sizes. In this paper, we introduce a new empirical method, which can estimate high tail index parameters well and might also be useful for relatively small sample sizes.