Maximum likelihood estimators for the extreme value index based on the block maxima method

TL;DR

This paper provides a theoretical foundation for using maximum likelihood estimators with the block maxima method to estimate the extreme value index, proving their existence and consistency under first order conditions.

Contribution

It establishes the existence and consistency of maximum likelihood estimators for the extreme value index based on block maxima, under minimal assumptions.

Findings

Proves the existence of MLE for the extreme value index.

Demonstrates the consistency of these estimators.

Provides a theoretical basis for practical extreme value analysis.

Abstract

The maximum likelihood method offers a standard way to estimate the three parameters of a generalized extreme value (GEV) distribution. Combined with the block maxima method, it is often used in practice to assess the extreme value index and normalization constants of a distribution satisfying a first order extreme value condition, assuming implicitely that the block maxima are exactly GEV distributed. This is unsatisfactory since the GEV distribution is a good approximation of the block maxima distribution only for blocks of large size. The purpose of this paper is to provide a theoretical basis for this methodology. Under a first order extreme value condition only, we prove the existence and consistency of the maximum likelihood estimators for the extreme value index and normalization constants within the framework of the block maxima method.