On the maximum likelihood estimator for the Generalized Extreme-Value distribution

TL;DR

This paper provides a formal proof of the asymptotic normality of the maximum likelihood estimator for the GEV distribution, addressing a longstanding theoretical gap in extreme-value analysis.

Contribution

It establishes the asymptotic normality of the MLE for the GEV distribution, including the first proof of its differentiability in quadratic mean.

Findings

Proof of asymptotic normality of GEV MLE

Analysis of differentiability in quadratic mean for GEV

Clarification of theoretical properties of GEV MLE

Abstract

The vanilla method in univariate extreme-value theory consists of fitting the three-parameter Generalized Extreme-Value (GEV) distribution to a sample of block maxima. Despite claims to the contrary, the asymptotic normality of the maximum likelihood estimator has never been established. In this paper, a formal proof is given using a general result on the maximum likelihood estimator for parametric families that are differentiable in quadratic mean but whose supports depend on the parameter. An interesting side result concerns the (lack of) differentiability in quadratic mean of the GEV family.