Gumbel Distribution
المؤلف:
Gumbel, E. J.
المصدر:
"Multivariate Extremal Distributions." Bull. Inst. Internat. de Statistique 37
الجزء والصفحة:
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6-4-2021
2405
Gumbel Distribution
There are essentially three types of Fisher-Tippett extreme value distributions. The most common is the type I distribution, which are sometimes referred to as Gumbel types or just Gumbel distributions. These are distributions of an extreme order statistic for a distribution of 
elements 
. In this work, the term "Gumbel distribution" is used to refer to the distribution corresponding to a minimum extreme value distribution (i.e., the distribution of the minimum 
).
The Gumbel distribution with location parameter 
and scale parameter 
is implemented in the Wolfram Language as GumbelDistribution[alpha, beta].
It has probability density function and distribution function
The mean, variance, skewness, and kurtosis excess are
where 
is the Euler-Mascheroni constant and 
is Apéry's constant.
The distribution of 
taken from a continuous uniform distribution over the unit interval has probability function
 |
(7)
|
and distribution function
 |
(8)
|
The 
th raw moment is given by
 |
(9)
|
The first few central moments are
The mean, variance, skewness, and kurtosis excess are therefore given by
If 
are instead taken from a standard normal distribution, then the corresponding cumulative distribution is
where 
is the normal distribution function. The probability distribution of 
is then
The mean 
and variance 
are expressible in closed form for small 
,
and
No exact expression is known for 
or 
, but there is an equation connecting them
 |
(31)
|
REFERENCES:
Gumbel, E. J. "Multivariate Extremal Distributions." Bull. Inst. Internat. de Statistique 37, 471-475, 1960a.
Gumbel, E. J. "Distributions del valeurs extremes en plusieurs dimensions." Publ. l'Inst. de Statistique, Paris 9, 171-173, 1960b.
Gumbel, E. J. "Bivariate Logistic Distributions." J. Amer. Stat. Assoc. 56, 335-349, 1961.
Gumbel, E. J. and Mustafi, C. K. "Some Analytical Properties of Bivariate Extreme Distributions." J. Amer. Stat. Assoc. 62, 569-588, 1967.
Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous Univariate Distributions, Vol. 2, 2nd ed. New York: Wiley, 1995.
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