Exponentiated Generalized Pareto Distribution: Properties and applications towards Extreme Value Theory

TL;DR

This paper introduces the Exponentiated Generalized Pareto Distribution (exGPD), derived from the GPD via log-transform, providing new tools for tail analysis and a practical plot for identifying tail indices in heavy-tailed data.

Contribution

The paper proposes the exGPD, derives its properties, and develops a novel tail index estimation plot, enhancing analysis of heavy-tailed distributions.

Findings

The exGPD effectively models heavy tail phenomena.

The new plot accurately identifies tail indices in datasets.

Numerical tests confirm the method's practical utility.

Abstract

The Generalized Pareto Distribution (GPD) plays a central role in modelling heavy tail phenomena in many applications. Applying the GPD to actual datasets however is a non-trivial task. One common way suggested in the literature to investigate the tail behaviour is to take logarithm to the original dataset in order to reduce the sample variability. Inspired by this, we propose and study the Exponentiated Generalized Pareto Distribution (exGPD), which is created via log-transform of the GPD variable. After introducing the exGPD we derive various distributional quantities, including the moment generating function, tail risk measures. As an application we also develop a plot as an alternative to the Hill plot to identify the tail index of heavy tailed datasets, based on the moment matching for the exGPD. Various numerical analyses with both simulated and actual datasets show that the proposed plot works well.