Analytical Conjugate Priors for Subclasses of Generalized Pareto Distributions

TL;DR

This paper derives analytical conjugate priors for specific subclasses of the Generalized Pareto distribution, facilitating Bayesian estimation of the distribution's support parameters, especially the lower bound.

Contribution

It introduces conjugate priors for four subclasses of the GP distribution, addressing a gap in Bayesian methods for support estimation, including the often-overlooked location parameter.

Findings

Conjugate priors are derived for Pareto, Shifted Exponential, Power, and Two-parameter Uniform subclasses.

Analytical solutions enable more efficient Bayesian inference for support parameters.

Some results are known, but the paper consolidates and extends the analytical framework.

Abstract

This article is written for pedagogical purposes aiming at practitioners trying to estimate the finite support of continuous probability distributions, i.e., the minimum and the maximum of a distribution defined on a finite domain. Generalized Pareto distribution GP({\theta}, {\sigma}, {\xi}) is a three-parameter distribution which plays a key role in Peaks-Over-Threshold framework for tail estimation in Extreme Value Theory. Estimators for GP often lack analytical solutions and the best known Bayesian methods for GP involves numerical methods. Moreover, existing literature focuses on estimating the scale {\sigma} and the shape {\xi}, lacking discussion of the estimation of the location {\theta} which is the lower support of (minimum value possible in) a GP. To fill the gap, we analyze four two-parameter subclasses of GP whose conjugate priors can be obtained analytically, although some of the results are known. Namely, we prove the conjugacy for {\xi} > 0 (Pareto), {\xi} = 0 (Shifted Exponential), {\xi} < 0 (Power), and {\xi} = -1 (Two-parameter Uniform).