Definitions

# Fréchet distribution

{{Probability distribution |
` name       =Fréchet|`
` type       =density|`
` pdf_image  =|`
` cdf_image  =|`
` parameters =$alpha in \left(0,infty\right]$ shape|`
` support    =$x>0$|`
pdf =$alpha ; x^\left\{-1-alpha\right\} ; e^\left\{-x^\left\{-alpha\right\}\right\}$| cdf =$e^\left\{-x^\left\{-alpha\right\}\right\}$| mean =$Gammaleft\left(1-frac\left\{1\right\}\left\{alpha\right\}right\right) text\left\{ if \right\} alpha>1$| median =$left\left(frac\left\{1\right\}\left\{log_e\left(2\right)\right\}right\right)^\left\{1/alpha\right\}$| mode =$left\left(frac\left\{alpha\right\}\left\{1+alpha\right\}right\right)^\left\{1/alpha\right\}$| variance =$Gammaleft\left(1-frac\left\{2\right\}\left\{alpha\right\}right\right)- left\left(Gammaleft\left(1-frac\left\{1\right\}\left\{alpha\right\}right\right)right\right)^2text\left\{ if \right\} alpha>2$|
` skewness   =|`
` g_k        =|`
` kurtosis   =|`
` entropy    =|`
` mgf        =|`
` char       =|`
}}

The Fréchet distribution is a special case of the generalized extreme value distribution. It has the cumulative probability function

where α>0 is a shape parameter. It can be generalised to include a location parameter m and a scale parameter s>0 with the cumulative probability function

Named for Maurice Fréchet who wrote a related paper in 1927, further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958

## Publications

• Fréchet, M., (1927). "Sur la loi de probabilité de l'écart maximum." Ann. Soc. Polon. Math. 6, 93.
• Fisher, R.A., Tippett, L.H.C., (1928). "Limiting forms of the frequency distribution of the largest and smallest member of a sample." Proc. Cambridge Philosophical Society 24:180-190.
• Gumbel, E.J. (1958). "Statistics of Extremes." Columbia University Press, New York.

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