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Extreme Value 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.

The Fisher-Tippett distribution corresponding to a maximum extreme value distribution (i.e., the distribution of the maximum X^(<N>)), sometimes known as the log-Weibull distribution, with location parameter alpha and scale parameter beta is implemented in the Wolfram Language as ExtremeValueDistribution[alpha, beta].

FisherTippettDistribution

It has probability density function and distribution function

P(x)=(e^((a-x)/b-e^((a-x)/b)))/b
(1)
D(x)=e^(-e^((a-x)/b)).
(2)

The moments can be computed directly by defining

z=exp((a-x)/b)
(3)
x=a-blnz
(4)
dz=-1/bexp((a-x)/b)dx.
(5)

Then the raw moments are

mu_n^'=int_(-infty)^inftyx^nP(x)dx
(6)
=1/bint_(-infty)^inftyx^nexp((a-x)/b)exp[-e^((a-x)/b)]dx
(7)
=-int_infty^0(a-blnz)^ne^(-z)dz
(8)
=int_0^infty(a-blnz)^ne^(-z)dz
(9)
=sum_(k=0)^(n)(n; k)(-1)^ka^(n-k)b^kint_0^infty(lnz)^ke^(-z)dz
(10)
=sum_(k=0)^(n)(n; k)a^(n-k)b^kI(k),
(11)

where I(k) are Euler-Mascheroni integrals. Plugging in the Euler-Mascheroni integrals I(k) gives

mu_0^'=1
(12)
mu_1^'=a+bgamma
(13)
mu_2^'=(a+bgamma)^2+1/6pi^2b^2
(14)
mu_3^'=2zeta(3)b^3+1/2(a+bgamma)pi^2b^2+(a+bgamma)^3
(15)
mu_4^'=a^4+4a^3bgamma+6a^2b^2(gamma^2+1/6pi^2)+4ab^3[gamma^3+1/2gammapi^2+2zeta(3)]+b^4[gamma^4+gamma^2pi^2+3/(20)pi^4+8gammazeta(3)],
(16)

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

The corresponding central moments are therefore

mu_2=1/6b^2pi^2
(17)
mu_3=2zeta(3)b^3
(18)
mu_4=3/(20)b^4pi^4,
(19)

giving mean, variance, skewness, and kurtosis excess of

mu=a+bgamma
(20)
sigma^2=1/6pi^2b^2
(21)
gamma_1=(12sqrt(6)zeta(3))/(pi^3)
(22)
gamma_2=(12)/5.
(23)

The characteristic function is

phi(t)=Gamma(1-ibetat)e^(ialphat),
(24)

where Gamma(z) is the gamma function (Abramowitz and Stegun 1972, p. 930).

An analog to the central limit theorem states that the asymptotic normalized distribution of M_n satisfies one of the three distributions

D(y)=exp(-e^(-y))
(25)
D(y)={0 if y<=0; exp(-y^(-a)) if y>0
(26)
D(y)={exp[-(-y)^a] if y<=0; 1 if y>0,
(27)

also known as Gumbel-type, Fréchet-type, and Weibull-type distributions, respectively.

The distributions of -y are also extreme value distributions. The Gumbel-type distribution for -y is implemented in as GumbelDistribution[alpha, beta]. The Weibull-type distribution for -y is a Weibull distribution. The two-parameter Weibull distribution is implemented as WeibullDistribution[alpha, beta].

See also

Euler-Mascheroni Integrals, Gumbel Distribution, Order Statistic, Weibull Distribution

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.Balakrishnan, N. and Cohen, A. C. Order Statistics and Inference. New York: Academic Press, 1991.David, H. A. Order Statistics, 2nd ed. New York: Wiley, 1981.Finch, S. R. "Extreme Value Constants." §5.16 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 363-367, 2003.Gibbons, J. D. and Chakraborti, S. (Eds.). Nonparametric Statistical Inference, 3rd rev. ext. ed. New York: Dekker, 1992.Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous Univariate Distributions, Vol. 2, 2nd ed. New York: Wiley, 1995.Natrella, M. "Extreme Value Distributions." §8.1.6.3 in Engineering Statistics Handbook. NIST/SEMATECH, 2005. http://www.itl.nist.gov/div898/handbook/apr/section1/apr163.htm.

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Extreme Value Distribution

Cite this as:

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

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