A comparative review of generalizations of the extreme value distribution

The extreme value distribution, also known as the Gumbel distribution, is widely applied for extreme value analysis but has certain drawbacks in practice because it is a non heavy-tailed distribution and is characterized by constant skewness and kurtosis. Our goal is to present a literature review of the distributions that contain the extreme value distribution embedded in them and to identify those that have flexible skewness and kurtosis and those that are heavy-tailed. The generalizations of the extreme value 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.