  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.
