Maximum likelihood estimation for the Fr\'echet distribution based on block maxima extracted from a time series

TL;DR

This paper establishes the statistical properties of maximum likelihood estimators for the Fréchet distribution when using block maxima from time series, addressing practical assumptions and providing theoretical and simulation validation.

Contribution

It proves consistency and asymptotic normality of MLEs for the Fréchet distribution in block maxima analysis, including dependent data from stationary time series.

Findings

MLEs are consistent for Fréchet parameters in block maxima methods.

Asymptotic normality of MLEs is established under general conditions.

Simulation results support theoretical findings.

Abstract

The block maxima method in extreme-value analysis proceeds by fitting an extreme-value distribution to a sample of block maxima extracted from an observed stretch of a time series. The method is usually validated under two simplifying assumptions: the block maxima should be distributed according to an extreme-value distribution and the sample of block maxima should be independent. Both assumptions are only approximately true. For general triangular arrays of block maxima attracted to the Fr\'echet distribution, consistency and asymptotic normality is established for the maximum likelihood estimator of the parameters of the limiting Fr\'echet distribution. The results are specialized to the setting of block maxima extracted from a strictly stationary time series. The case where the underlying random variables are independent and identically distributed is further worked out in detail. The results are illustrated by theoretical examples and Monte Carlo simulations.