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Fréchet Distribution

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The Fréchet distribution with shape parameter alpha>0 and scale parameter s>0 has probability density function and distribution function

P(x)=alpha/s(x/s)^(-1-alpha)exp[-(x/s)^(-alpha)]
(1)
D(x)=exp[-(x/s)^(-alpha)],
(2)

for x>0, with P(x)=D(x)=0 for x<=0. It is implemented in the Wolfram Language as FrechetDistribution[alpha, s].

The nth raw moment is

mu_n^'=s^nGamma(1-n/alpha)
(3)

for n<alpha. In particular, the mean, variance, and median are

mu=sGamma(1-alpha^(-1))
(4)
sigma^2=s^2[Gamma(1-2alpha^(-1))-Gamma^2(1-alpha^(-1))]
(5)
mu_(1/2)=s/((ln2)^(1/alpha)).
(6)

The mean is finite for alpha>1, and the variance is finite for alpha>2.

The Fréchet distribution is heavy-tailed, with

1-D(x)∼(x/s)^(-alpha)
(7)

as x->infty. It is the type II extreme value distribution in the Fisher-Tippett-Gnedenko theorem and, up to location and scale, corresponds to a positive generalized extreme value shape parameter xi=1/alpha. The Pareto distribution and other distributions with regularly varying upper tails belong to its maximum domain of attraction (de Haan and Ferreira 2006).

If X has a Fréchet distribution with parameters alpha and s, then s/X has a Weibull distribution with shape parameter alpha and unit scale.

See also

Extreme Value Distribution, Extreme Value Theory, Fisher-Tippett Distribution, Fisher-Tippett-Gnedenko Theorem, Gumbel Distribution, Order Statistic, Pareto Distribution, Weibull Distribution

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References

de Haan, L. and Ferreira, A. Extreme Value Theory: An Introduction. New York: Springer-Verlag, 2006.Fréchet, M. "Sur la loi de probabilité de l'écart maximum." Ann. de la Soc. Polonaise de Math. 6, 93-116, 1927.Johnson, N.; Kotz, S.; and Balakrishnan, N. Continuous Univariate Distributions, Vol. 2, 2nd ed. New York: Wiley, 1995.

Cite this as:

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

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