Uniqueness and global optimality of the maximum likelihood estimator for the generalized extreme value distribution

TL;DR

This paper proves the global uniqueness and consistency of the maximum likelihood estimator for the generalized extreme value distribution, filling a gap in the theoretical understanding of this widely used model.

Contribution

It establishes the asymptotic properties of the MLE for the GEV distribution, including its global optimality and uniqueness, which were previously unproven.

Findings

MLE is globally unique for GEV distribution

Uniform consistency of limit relations near the shape parameter

Provides theoretical foundation for likelihood-based inference

Abstract

The three-parameter generalized extreme value distribution arises from classical univariate extreme value theory and is in common use for analyzing the far tail of observed phenomena. Curiously, important asymptotic properties of likelihood-based estimation under this standard model have yet to be established. In this paper, we formally prove that the maximum likelihood estimator is global and unique. An interesting secondary result entails the uniform consistency of a class of limit relations in a tight neighborhood of the shape parameter.