Generalized Gumbel distribution

Abstract

A generalization of the Gumbel distribution is presented to deal with general situations in modeling univariate data with broad range of skewness in the density function. This generalization is derived by considering a logarithmic transformation of an odd Weibull random variable. As a result, the generalized Gumbel distribution is not only useful for testing goodness-of-fit of Gumbel and reverse-Gumbel distributions as submodels, but it is also convenient for modeling and fitting a wide variety of data sets that are not possible to be modeled by well-known distributions. Skewness and kurtosis shapes of the generalized Gumbel distribution are illustrated by constructing the Galton’s skewness and Moor’s kurtosis plane. Parameters are estimated by using maximum likelihood method in two different ways due to the fact that the reverse transformation of the proposed distribution does not change its density function. In order to illustrate the flexibility of this generalization, wave and surge height data set is analyzed, and the fitness is compared with Gumbel and generalized extreme value distributions.

Keywords:

• coverage probabilities
• generalized extreme value distribution
• Gumbel distribution
• odd Weibull distribution
• skewness and kurtosis

Acknowledgements

Two unknown referees and the editor were very helpful in revising an earlier version.

Notes

^†This article is devoted to the celebration of applications of 50 years of Gumbel distribution, 20 years of Moor’s kurtosis, and 125 years of Galton’s skewness.