Maxima of Weibull-like distributions and the Lambert W function

TL;DR

This paper introduces improved methods for calculating norming constants of Weibull-like distributions, including Gamma and chi-squared, using Lambert W function asymptotics to better approximate the Gumbel law for maxima.

Contribution

It proposes novel asymptotic expressions for norming constants based on Lambert W function, enhancing accuracy over traditional methods for Weibull-like distributions.

Findings

Improved accuracy of norming constants for maxima of Weibull-like distributions.

Enhanced approximation of the Gumbel law for moderate and large samples.

Application to maxima of Gamma distributions in practical problems.

Abstract

The Weibull--like distributions form a large class of probability distributions that belong to the domain of attraction for the maxima of the Gumbel law. Besides the Weibull distribution, it includes important distributions as the Gamma laws and, in particular, the $\chi^2$ distributions. In order to have explicit expressions of the norming constants for the maxima it is necessary to solve asymptotically a nonlinear equation; however, for some members of that family, numerical and simulation studies show that the constants that are usual suggested are inaccurate for moderate or even large sample sizes. In this paper we propose other norming constants computed with the asymptotics of the Lambert W function that significantly improve the accuracy of the approximation to the Gumbel law. These results are applied to the computation of the constants for the maxima of Gamma random variables that appear in some applied problems.