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Gumbel Distribution

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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 N elements X_i. 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 X^(<1>)).

The Gumbel distribution with location parameter alpha and scale parameter beta is implemented in the Wolfram Language as GumbelDistribution[alpha, beta].

It has probability density function and distribution function

P(x)=1/betaexp[(x-alpha)/beta-exp((x-alpha)/beta)]
(1)
D(x)=1-exp[-exp((x-alpha)/beta)].
(2)

The mean, variance, skewness, and kurtosis excess are

mu=alpha-gammabeta
(3)
sigma^2=1/6pi^2beta^2
(4)
gamma_1=-(12sqrt(6)zeta(3))/(pi^3)
(5)
gamma_2=(12)/5,
(6)

where gamma is the Euler-Mascheroni constant and zeta(3) is Apéry's constant.

ExtremeValueDistribution

The distribution of X^(<1>) taken from a continuous uniform distribution over the unit interval has probability function

P_N(x)=Nx^(N-1),
(7)

and distribution function

D_N(x)=x^N.
(8)

The kth raw moment is given by

mu_k^'=N/(k+N).
(9)

The first few central moments are

mu_2=N/((N+1)^2(N+2))
(10)
mu_3=-(2N(N-1))/((N+1)^3(N^2+5N+6))
(11)
mu_4=(N(9N^2-3N+6))/((N+1)^4(N+2)(N+3)(N+4)).
(12)

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

mu=N/(N+1)
(13)
sigma^2=N/((N+1)^2(N+2))
(14)
gamma_1=-(2(N-1))/(N+3)sqrt((N+2)/N)
(15)
gamma_2=(6(N^3-N^2-6N+2))/(N(N+3)(N+4)).
(16)

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

F(x)=1/(sqrt(2pi))int_(-infty)^xe^(-t^2/2)dt
(17)
=1/2+Phi(x),
(18)

where Phi(x) is the normal distribution function. The probability distribution of X^(<1>) is then

P(M_n<x)=[F(x)]^n
(19)
=n/(sqrt(2pi))int_(-infty)^x[F(t)]^(n-1)e^(-t^2/2)dt.
(20)

The mean mu(n) and variance sigma^2(n) are expressible in closed form for small n,

mu(1)=0
(21)
mu(2)=1/(sqrt(pi))
(22)
mu(3)=3/(2sqrt(pi))
(23)
mu(4)=3/(2sqrt(pi))[1+2/pisin^(-1)(1/3)]
(24)
mu(5)=5/(4sqrt(pi))[1+6/pisin^(-1)(1/3)]
(25)

and

sigma^2(1)=1
(26)
sigma^2(2)=1-1/pi
(27)
sigma^2(3)=(4pi-9+2sqrt(3))/(4pi)
(28)
sigma^2(4)=1+(sqrt(3))/pi-mu^2(4)
(29)
sigma^2(5)=1+(5sqrt(3))/(4pi)+(5sqrt(3))/(2pi^2)sin^(-1)(1/4)-mu^2(5).
(30)

No exact expression is known for mu(6) or sigma^2(6), but there is an equation connecting them

mu^2(6)+sigma^2(6)=1+(5sqrt(3))/(4pi)+(15sqrt(3))/(2pi^2)sin^(-1)(1/4).
(31)

See also

Extreme Value Distribution, Inverse Gumbel Distribution

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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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Gumbel Distribution

Cite this as:

Weisstein, Eric W. "Gumbel Distribution." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GumbelDistribution.html

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