# Maximum likelihood estimators based on the block maxima method

## TL;DR

This paper develops asymptotic theory for maximum likelihood estimators based on the block maxima method in extreme value analysis, providing insights into their distribution, bias, and comparison with other estimators.

## Contribution

It introduces the asymptotic normality of MLEs for block maxima, including bias analysis, and compares their efficiency with other semi-parametric estimators in EVT.

## Key findings

- MLEs are asymptotically normal with bias depending on the tail index and second order parameter.
- The paper compares asymptotic variances and biases of MLE and other estimators in EVT.
- Results facilitate optimal estimator selection based on mean square error.

## Abstract

The extreme value index is a fundamental parameter in univariate Extreme Value Theory (EVT). It captures the tail behavior of a distribution and is central in the extrapolation beyond observed data. Among other semi-parametric methods (such as the popular Hill's estimator), the Block Maxima (BM) and Peaks-Over-Threshold (POT) methods are widely used for assessing the extreme value index and related normalizing constants. We provide asymptotic theory for the maximum likelihood estimators (MLE) based on the BM method. Our main result is the asymptotic normality of the MLE with a non-trivial bias depending on the extreme value index and on the so-called second order parameter. Our approach combines asymptotic expansions of the likelihood process and of the empirical quantile process of block maxima. The results permit to complete the comparison of most common semi-parametric estimators in EVT (MLE and probability weighted moment estimators based on the POT or BM methods) through their asymptotic variances, biases and optimal mean square errors.