Accurately approximating extreme value statistics

TL;DR

This paper develops a new theoretical approach to accurately approximate the distribution of extreme values for large samples, improving upon the classical Gumbel limit by using power series and transformations, with applications to unknown distributions.

Contribution

The authors introduce a method that refines the Gumbel approximation for extreme value distributions using power series and transformations, enhancing accuracy for large samples.

Findings

Provides a power series representation of scale and width parameters.

Derives functional corrections to the Gumbel limit via Taylor expansion.

Improves large deviation descriptions and characterizes unknown distributions.

Abstract

We consider the extreme value statistics of $N$ independent and identically distributed random variables, which is a classic problem in probability theory. When $N\to\infty$, fluctuations around the maximum of the variables are described by the Fisher-Tippett-Gnedenko theorem, which states that the distribution of maxima converges to one out of three limiting forms. Among these is the Gumbel distribution, for which the convergence rate with $N$ is of a logarithmic nature. Here, we present a theory that allows one to use the Gumbel limit to accurately approximate the exact extreme value distribution. We do so by representing the scale and width parameters as power series, and by a transformation of the underlying distribution. We consider functional corrections to the Gumbel limit as well, showing they are obtainable via Taylor expansion. Our method also improves the description of large deviations from the mean extreme value. Additionally, it helps to characterize the extreme value statistics when the underlying distribution is unknown, for example when fitting experimental data.