Asymptotic posterior normality of the generalized extreme value distribution

TL;DR

This paper proves that the Bayesian posterior distribution of the GEV parameters converges to a normal distribution as sample size increases, despite the distribution's complex support and regularity issues.

Contribution

It provides the first rigorous proof of asymptotic normality for the GEV posterior, addressing a key theoretical gap in extreme value Bayesian analysis.

Findings

Posterior distribution converges to a normal distribution asymptotically.

Analysis involves complex integrals over the GEV likelihood with parameter-dependent support.

Addresses regularity condition challenges in GEV asymptotics.

Abstract

The univariate generalized extreme value (GEV) distribution is the most commonly used tool for analyzing the properties of rare events. The ever greater utilization of Bayesian methods for extreme value analysis warrants detailed theoretical investigation, which has thus far been underdeveloped. Even the most basic asymptotic results are difficult to obtain because the GEV fails to satisfy standard regularity conditions. Here, we prove that the posterior distribution of the GEV parameter vector, given $n$ independent and identically distributed samples, converges in distribution to a trivariate normal distribution. The proof necessitates analyzing integrals of the GEV likelihood function over the entire parameter space, which requires considerable care because the support of the GEV density depends on the parameters in complicated ways.