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
The moments can be computed directly by defining
Then the raw moments are
where 
are Euler-Mascheroni integrals. Plugging
in the Euler-Mascheroni integrals 
gives
where 
is the Euler-Mascheroni constant and
is Apéry's constant.
The corresponding central moments are therefore
giving mean, variance, skewness, and kurtosis excess of
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
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].
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.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Extreme Value Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/ExtremeValueDistribution.html