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The generalized extreme value distribution and its particular case, theGumbel extreme value distribution, are widely applied for extreme valueanalysis. The Gumbel distribution has certain drawbacks because it is anon-heavy-tailed distribution and is characterized by constant skewness andkurtosis. The generalized extreme value distribution is frequently used in thiscontext because it encompasses the three possible limiting distributions for anormalized maximum of infinite samples of independent and identicallydistributed observations. However, the generalized extreme value distributionmight not be a suitable model when each observed maximum does not come from alarge number of observations. Hence, other forms of generalizations of theGumbel distribution might be preferable. Our goal is to collect in the presentliterature the distributions that contain the Gumbel distribution embedded inthem and to identify those that have flexible skewness and kurtosis, areheavy-tailed and could be competitive with the generalized extreme valuedistribution. The generalizations of the Gumbel distribution are described andcompared using an application to a wind speed data set and Monte Carlosimulations. We show that some distributions suffer from overparameterizationand coincide with other generalized Gumbel distributions with a smaller numberof parameters, i.e., are non-identifiable. Our study suggests that thegeneralized extreme value distribution and a mixture of two extreme valuedistributions should be considered in practical applications.

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