# Fréchet distribution

The Fréchet distribution, also known as inverse Weibull distribution,[2][3] is a special case of the generalized extreme value distribution. It has the cumulative distribution function

${\displaystyle \Pr(X\leq x)=e^{-x^{-\alpha }}{\text{ if }}x>0.}$
Parameters Probability density function Cumulative distribution function ${\displaystyle \alpha \in (0,\infty )}$ shape. (Optionally, two more parameters) ${\displaystyle s\in (0,\infty )}$ scale (default: ${\displaystyle s=1\,}$ ) ${\displaystyle m\in (-\infty ,\infty )}$ location of minimum (default: ${\displaystyle m=0\,}$ ) ${\displaystyle x>m}$ ${\displaystyle {\frac {\alpha }{s}}\;\left({\frac {x-m}{s}}\right)^{-1-\alpha }\;e^{-({\frac {x-m}{s}})^{-\alpha }}}$ ${\displaystyle e^{-({\frac {x-m}{s}})^{-\alpha }}}$ ${\displaystyle {\begin{cases}\ m+s\Gamma \left(1-{\frac {1}{\alpha }}\right)&{\text{for }}\alpha >1\\\ \infty &{\text{otherwise}}\end{cases}}}$ ${\displaystyle m+{\frac {s}{\sqrt[{\alpha }]{\log _{e}(2)}}}}$ ${\displaystyle m+s\left({\frac {\alpha }{1+\alpha }}\right)^{1/\alpha }}$ ${\displaystyle {\begin{cases}\ s^{2}\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\left(\Gamma \left(1-{\frac {1}{\alpha }}\right)\right)^{2}\right)&{\text{for }}\alpha >2\\\ \infty &{\text{otherwise}}\end{cases}}}$ ${\displaystyle {\begin{cases}\ {\frac {\Gamma \left(1-{\frac {3}{\alpha }}\right)-3\Gamma \left(1-{\frac {2}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+2\Gamma ^{3}\left(1-{\frac {1}{\alpha }}\right)}{\sqrt {\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right)^{3}}}}&{\text{for }}\alpha >3\\\ \infty &{\text{otherwise}}\end{cases}}}$ ${\displaystyle {\begin{cases}\ -6+{\frac {\Gamma \left(1-{\frac {4}{\alpha }}\right)-4\Gamma \left(1-{\frac {3}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+3\Gamma ^{2}\left(1-{\frac {2}{\alpha }}\right)}{\left[\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right]^{2}}}&{\text{for }}\alpha >4\\\ \infty &{\text{otherwise}}\end{cases}}}$ ${\displaystyle 1+{\frac {\gamma }{\alpha }}+\gamma +\ln \left({\frac {s}{\alpha }}\right)}$ , where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant. [1] Note: Moment ${\displaystyle k}$ exists if ${\displaystyle \alpha >k}$ [1]

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

${\displaystyle \Pr(X\leq x)=e^{-\left({\frac {x-m}{s}}\right)^{-\alpha }}{\text{ if }}x>m.}$

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.

## Characteristics

The single parameter Fréchet with parameter ${\displaystyle \alpha }$ has standardized moment

${\displaystyle \mu _{k}=\int _{0}^{\infty }x^{k}f(x)dx=\int _{0}^{\infty }t^{-{\frac {k}{\alpha }}}e^{-t}\,dt,}$

(with ${\displaystyle t=x^{-\alpha }}$ ) defined only for ${\displaystyle k<\alpha }$ :

${\displaystyle \mu _{k}=\Gamma \left(1-{\frac {k}{\alpha }}\right)}$

where ${\displaystyle \Gamma \left(z\right)}$ is the Gamma function.

In particular:

• For ${\displaystyle \alpha >1}$ the expectation is ${\displaystyle E[X]=\Gamma (1-{\tfrac {1}{\alpha }})}$
• For ${\displaystyle \alpha >2}$ the variance is ${\displaystyle {\text{Var}}(X)=\Gamma (1-{\tfrac {2}{\alpha }})-{\big (}\Gamma (1-{\tfrac {1}{\alpha }}){\big )}^{2}.}$

The quantile ${\displaystyle q_{y}}$ of order ${\displaystyle y}$ can be expressed through the inverse of the distribution,

${\displaystyle q_{y}=F^{-1}(y)=\left(-\log _{e}y\right)^{-{\frac {1}{\alpha }}}}$ .

In particular the median is:

${\displaystyle q_{1/2}=(\log _{e}2)^{-{\frac {1}{\alpha }}}.}$

The mode of the distribution is ${\displaystyle \left({\frac {\alpha }{\alpha +1}}\right)^{\frac {1}{\alpha }}.}$

Especially for the 3-parameter Fréchet, the first quartile is ${\displaystyle q_{1}=m+{\frac {s}{\sqrt[{\alpha }]{\log(4)}}}}$ and the third quartile ${\displaystyle q_{3}=m+{\frac {s}{\sqrt[{\alpha }]{\log({\frac {4}{3}})}}}.}$

Also the quantiles for the mean and mode are:

${\displaystyle F(mean)=\exp \left(-\Gamma ^{-\alpha }\left(1-{\frac {1}{\alpha }}\right)\right)}$
${\displaystyle F(mode)=\exp \left(-{\frac {\alpha +1}{\alpha }}\right).}$

## Applications

However, in most hydrological applications, the distribution fitting is via the generalized extreme value distribution as this avoids imposing the assumption that the distribution does not have a lower bound (as required by the Frechet distribution).

• One test to assess whether a multivariate distribution is asymptotically dependent or independent consists of transforming the data into standard Fréchet margins using the transformation ${\displaystyle Z_{i}=-1/\log F_{i}(X_{i})}$ and then mapping from Cartesian to pseudo-polar coordinates ${\displaystyle (R,W)=(Z_{1}+Z_{2},Z_{1}/(Z_{1}+Z_{2}))}$ . Values of ${\displaystyle R\gg 1}$ correspond to the extreme data for which at least one component is large while ${\displaystyle W}$ approximately 1 or 0 corresponds to only one component being extreme.
• If ${\displaystyle X\sim U(0,1)\,}$ (Uniform distribution (continuous)) then ${\displaystyle m+s(-\log(X))^{-1/\alpha }\sim {\textrm {Frechet}}(\alpha ,s,m)\,}$
• If ${\displaystyle X\sim {\textrm {Frechet}}(\alpha ,s,m)\,}$ then ${\displaystyle kX+b\sim {\textrm {Frechet}}(\alpha ,ks,km+b)\,}$
• If ${\displaystyle X_{i}\sim {\textrm {Frechet}}(\alpha ,s,m)\,}$ and ${\displaystyle Y=\max\{\,X_{1},\ldots ,X_{n}\,\}\,}$ then ${\displaystyle Y\sim {\textrm {Frechet}}(\alpha ,n^{\tfrac {1}{\alpha }}s,m)\,}$
• The cumulative distribution function of the Frechet distribution solves the maximum stability postulate equation
• If ${\displaystyle X\sim {\textrm {Frechet}}(\alpha ,s,m=0)\,}$ then its reciprocal is Weibull-distributed: ${\displaystyle X^{-1}\sim {\textrm {Weibull}}(k=\alpha ,\lambda =s^{-1})\,}$

## References

1. Muraleedharan. G, C. Guedes Soares and Cláudia Lucas (2011). "Characteristic and Moment Generating Functions of Generalised Extreme Value Distribution (GEV)". In Linda. L. Wright (Ed.), Sea Level Rise, Coastal Engineering, Shorelines and Tides, Chapter 14, pp. 269–276. Nova Science Publishers. ISBN 978-1-61728-655-1
2. Khan M.S.; Pasha G.R.; Pasha A.H. (February 2008). "Theoretical Analysis of Inverse Weibull Distribution" (PDF). WSEAS TRANSACTIONS on MATHEMATICS. 7 (2). pp. 30–38.
3. de Gusmão, FelipeR.S. and Ortega, EdwinM.M. and Cordeiro, GaussM. (2011). "The generalized inverse Weibull distribution". Statistical Papers. 52 (3). Springer-Verlag. pp. 591–619. doi:10.1007/s00362-009-0271-3. ISSN 0932-5026.CS1 maint: uses authors parameter (link)
4. Coles, Stuart (2001). An Introduction to Statistical Modeling of Extreme Values. Springer-Verlag. ISBN 978-1-85233-459-8.

## 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 24:180190.
• Gumbel, E.J. (1958). "Statistics of Extremes." Columbia University Press, New York.
• Kotz, S.; Nadarajah, S. (2000) Extreme value distributions: theory and applications, World Scientific. ISBN 1-86094-224-5