A comparative review of generalizations of the Gumbel extreme value distribution with an application to wind speed data

TL;DR

This paper reviews various generalizations of the Gumbel distribution, comparing their flexibility and suitability for extreme value analysis, especially in wind speed data, highlighting issues like overparameterization and non-identifiability.

Contribution

It systematically compares Gumbel distribution generalizations, identifying those with flexible skewness, kurtosis, and heavy tails, and evaluates their practical applicability.

Findings

Some distributions are overparameterized and non-identifiable.

The generalized extreme value distribution is often suitable for practical use.

Mixture models of extreme value distributions can be effective.

Abstract

The generalized extreme value distribution and its particular case, the Gumbel extreme value distribution, are widely applied for extreme value analysis. The Gumbel distribution has certain drawbacks because it is a non-heavy-tailed distribution and is characterized by constant skewness and kurtosis. The generalized extreme value distribution is frequently used in this context because it encompasses the three possible limiting distributions for a normalized maximum of infinite samples of independent and identically distributed observations. However, the generalized extreme value distribution might not be a suitable model when each observed maximum does not come from a large number of observations. Hence, other forms of generalizations of the Gumbel distribution might be preferable. Our goal is to collect in the present literature the distributions that contain the Gumbel distribution embedded in them and to identify those that have flexible skewness and kurtosis, are heavy-tailed and could be competitive with the generalized extreme value distribution. The generalizations of the Gumbel distribution are described and compared using an application to a wind speed data set and Monte Carlo simulations. We show that some distributions suffer from overparameterization and coincide with other generalized Gumbel distributions with a smaller number of parameters, i.e., are non-identifiable. Our study suggests that the generalized extreme value distribution and a mixture of two extreme value distributions should be considered in practical applications.