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

			(1)
			(2)

The mean, variance, skewness, and kurtosis excess are

			(3)
			(4)
			(5)
			(6)

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

			(10)
			(11)
			(12)

The mean, variance, skewness, and kurtosis excess are therefore given by

			(13)
			(14)
			(15)
			(16)

If  are instead taken from a standard normal distribution, then the corresponding cumulative distribution is

			(17)
			(18)

where  is the normal distribution function. The probability distribution of  is then

			(19)
			(20)

The mean  and variance  are expressible in closed form for small ,

			(21)
			(22)
			(23)
			(24)
			(25)

and

			(26)
			(27)
			(28)
			(29)
			(30)

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.