An entropic view of Pickands' theorem

TL;DR

This paper presents an entropic perspective on Pickands' theorem, demonstrating that the distribution of normalized excesses converges to a Tsallis maximum entropy distribution, linking generalized Pareto distributions to entropy principles.

Contribution

It introduces an entropic framework for understanding the convergence to GPD in tail modeling, connecting Tsallis entropy to extreme value theory.

Findings

Normalized excess distributions converge to Tsallis maximum entropy solutions.

GPD distributions are shown to be related to Renyi-Tsallis maximum entropy.

Highlights the importance of Tsallis distributions in practical tail modeling.

Abstract

It is shown that distributions arising in Renyi-Tsallis maximum entropy setting are related to the Generalized Pareto Distributions (GPD) that are widely used for modeling the tails of distributions. The relevance of such modelization, as well as the ubiquity of GPD in practical situations follows from Balkema-De Haan-Pickands theorem on the distribution of excesses (over a high threshold). We provide an entropic view of this result, by showing that the distribution of a suitably normalized excess variable converges to the solution of a maximum Tsallis entropy, which is the GPD. This highlights the relevance of the so-called Tsallis distributions in many applications as well as some relevance to the use of the corresponding entropy.