Bayesian Estimation of the Threshold of a Generalised Pareto Distribution for Heavy-Tailed Observations

TL;DR

This paper introduces Bayesian methods for estimating the threshold of a generalized Pareto distribution tailored for heavy-tailed data, using priors based on order statistics, and evaluates their performance through simulations and real-world applications.

Contribution

It proposes two novel prior distributions for the threshold parameter, based on order statistics, enhancing Bayesian estimation in heavy-tailed distribution modeling.

Findings

Uniform prior performs well in simulations.

Worth-based prior adapts to data characteristics.

Both priors are effective in insurance and finance applications.

Abstract

In this paper, we discuss a method to define prior distributions for the threshold of a generalised Pareto distribution, in particular when its applications are directed to heavy-tailed data. We propose to assign prior probabilities to the order statistics of a given set of observations. In other words, we assume that the threshold coincides to one of the data points. We show two ways of defining a prior: by assigning equal mass to each order statistic, that is a uniform prior, and by considering the worth that every order statistic has in representing the true threshold. Both proposed priors represent a scenario of minimal information, and we study their adequacy through simulation exercises and by analysing two applications from insurance and from finance.