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

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, that is, 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.

KEYWORDS:

• Generalized extreme value distribution
• Gumbel distribution
• heavy-tailed distribution
• non-identifiable model
• kurtosis
• wind speed

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

We gratefully acknowledge financial support from the Brazilian agencies CNPq, CAPES, and FAPESP.

Notes

1. A family of distributions with probability density function (pdf) $f(x;\theta ),\theta \in \mathrm{\Theta }$, is said to be identifiable if, for any θ and ${\theta }^{\ast }$ in the parameter space Θ, $f(x;\theta )=f(x;{\theta }^{\ast })\iff \theta ={\theta }^{\ast }$.

2. The parameterization for the gamma distribution is such that, if $W\sim \mathrm{gamma}(\alpha ,\beta )$, its pdf is $f\left(w\right)=\left({\beta }^{\alpha }/\mathrm{\Gamma }\right(\alpha \left)\right)w^{\alpha -1}\exp (-\beta w)$, $w>0$.

3. Cooray [31] considers the parameter space $\mu \in \mathbb{R}$ and $0<\alpha \sigma <\mathrm{\infty }$.

4. If $W\sim \mathrm{F}(a,b)$ its pdf is $f\left(w\right)=\left(1/B\right(a,b\left)\right)\left(a/b{)}^a{w}^{a-1}/\right(1+\left(a/b\right)w{)}^{(a+b)}$, for $w>0$.

5. For purposes of parameter estimation, we follow Aitkin and Rubin,[48] who suggest theoretical parameter constraints but no parameters constraints for estimation.

6. Recall that we restrict our attention to the identifiable family of distributions only.

7. More specifically, http://www.ncdc.noaa.gov/ $\to$ I want to search for data at a particular location. $\to$ Additional Data Access: Publications $\to$ Local Climatological Data (last accessed on 23 January 2014)