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 .

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

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 are Euler-Mascheroni integrals. Plugging in the Euler-Mascheroni integrals gives

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

where is the Euler-Mascheroni constant and 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 is the gamma function (Abramowitz and Stegun 1972, p. 930).

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

			(25)
			(26)
			(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].

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