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Extreme Value Distribution

03:43 PM

5 / 4 / 2021

Author : Abramowitz, M. and Stegun, I. A.

Book or Source : Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.

page : ...

Extreme Value 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 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

			(1)
			(2)

The moments can be computed directly by defining

			(3)
			(4)
			(5)

Then the raw moments are

			(6)
			(7)
			(8)
			(9)
			(10)
			(11)

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

			(12)
			(13)
			(14)
			(15)
			(16)

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

The corresponding central moments are therefore

			(17)
			(18)
			(19)

giving mean, variance, skewness, and kurtosis excess of

			(20)
			(21)
			(22)
			(23)

The characteristic function is

(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

			(25)
		$<span style='font-family: rasol;'> { </span>0 if y<=0; exp(-y^(-a)) if y>0$	(26)
		$<span style='font-family: rasol;'> { </span>exp[-(-y)^a] if y<=0; 1 if y>0,$	(27)

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

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

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. https://www.itl.nist.gov/div898/handbook/apr/section1/apr163.htm.

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Extreme Value Distribution