# Fréchet distribution

Parameters Probability density function Cumulative distribution function $\alpha \in (0,\infty )$ shape. (Optionally, two more parameters) $s\in (0,\infty )$ scale (default: $s=1\,$ ) $m\in (-\infty ,\infty )$ location of minimum (default: $m=0\,$ ) $x>m$ ${\frac {\alpha }{s}}\;\left({\frac {x-m}{s}}\right)^{-1-\alpha }\;e^{-({\frac {x-m}{s}})^{-\alpha }}$ $e^{-({\frac {x-m}{s}})^{-\alpha }}$ ${\begin{cases}\ m+s\Gamma \left(1-{\frac {1}{\alpha }}\right)&{\text{for }}\alpha >1\\\ \infty &{\text{otherwise}}\end{cases}}$ $m+{\frac {s}{\sqrt[{\alpha }]{\log _{e}(2)}}}$ $m+s\left({\frac {\alpha }{1+\alpha }}\right)^{1/\alpha }$ ${\begin{cases}\ s^{2}\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\left(\Gamma \left(1-{\frac {1}{\alpha }}\right)\right)^{2}\right)&{\text{for }}\alpha >2\\\ \infty &{\text{otherwise}}\end{cases}}$ ${\begin{cases}\ {\frac {\Gamma \left(1-{\frac {3}{\alpha }}\right)-3\Gamma \left(1-{\frac {2}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+2\Gamma ^{3}\left(1-{\frac {1}{\alpha }}\right)}{\sqrt {\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right)^{3}}}}&{\text{for }}\alpha >3\\\ \infty &{\text{otherwise}}\end{cases}}$ ${\begin{cases}\ -6+{\frac {\Gamma \left(1-{\frac {4}{\alpha }}\right)-4\Gamma \left(1-{\frac {3}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+3\Gamma ^{2}\left(1-{\frac {2}{\alpha }}\right)}{\left[\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right]^{2}}}&{\text{for }}\alpha >4\\\ \infty &{\text{otherwise}}\end{cases}}$ $1+{\frac {\gamma }{\alpha }}+\gamma +\ln \left({\frac {s}{\alpha }}\right)$ , where $\gamma$ is the Euler–Mascheroni constant.  Note: Moment $k$ exists if $\alpha >k$ The Fréchet distribution, also known as inverse Weibull distribution, is a special case of the generalized extreme value distribution. It has the cumulative distribution function

$\Pr(X\leq x)=e^{-x^{-\alpha }}{\text{ if }}x>0.$ 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

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

$\mu _{k}=\int _{0}^{\infty }x^{k}f(x)dx=\int _{0}^{\infty }t^{-{\frac {k}{\alpha }}}e^{-t}\,dt,$ (with $t=x^{-\alpha }$ ) defined only for $k<\alpha$ :

$\mu _{k}=\Gamma \left(1-{\frac {k}{\alpha }}\right)$ where $\Gamma \left(z\right)$ is the Gamma function.

In particular:

• For $\alpha >1$ the expectation is $E[X]=\Gamma (1-{\tfrac {1}{\alpha }})$ • For $\alpha >2$ the variance is ${\text{Var}}(X)=\Gamma (1-{\tfrac {2}{\alpha }})-{\big (}\Gamma (1-{\tfrac {1}{\alpha }}){\big )}^{2}.$ The quantile $q_{y}$ of order $y$ can be expressed through the inverse of the distribution,

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

In particular the median is:

$q_{1/2}=(\log _{e}2)^{-{\frac {1}{\alpha }}}.$ The mode of the distribution is $\left({\frac {\alpha }{\alpha +1}}\right)^{\frac {1}{\alpha }}.$ Especially for the 3-parameter Fréchet, the first quartile is $q_{1}=m+{\frac {s}{\sqrt[{\alpha }]{\log(4)}}}$ and the third quartile $q_{3}=m+{\frac {s}{\sqrt[{\alpha }]{\log({\frac {4}{3}})}}}.$ Also the quantiles for the mean and mode are:

$F(mean)=\exp \left(-\Gamma ^{-\alpha }\left(1-{\frac {1}{\alpha }}\right)\right)$ $F(mode)=\exp \left(-{\frac {\alpha +1}{\alpha }}\right).$ ## 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).[citation needed]

• 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 $Z_{i}=-1/\log F_{i}(X_{i})$ and then mapping from Cartesian to pseudo-polar coordinates $(R,W)=(Z_{1}+Z_{2},Z_{1}/(Z_{1}+Z_{2}))$ . Values of $R\gg 1$ correspond to the extreme data for which at least one component is large while $W$ approximately 1 or 0 corresponds to only one component being extreme.

## Related distributions

• If $X\sim U(0,1)\,$ (Uniform distribution (continuous)) then $m+s(-\log(X))^{-1/\alpha }\sim {\textrm {Frechet}}(\alpha ,s,m)\,$ • If $X\sim {\textrm {Frechet}}(\alpha ,s,m)\,$ then $kX+b\sim {\textrm {Frechet}}(\alpha ,ks,km+b)\,$ • If $X_{i}\sim {\textrm {Frechet}}(\alpha ,s,m)\,$ and $Y=\max\{\,X_{1},\ldots ,X_{n}\,\}\,$ then $Y\sim {\textrm {Frechet}}(\alpha ,n^{\tfrac {1}{\alpha }}s,m)\,$ • The cumulative distribution function of the Frechet distribution solves the maximum stability postulate equation
• If $X\sim {\textrm {Frechet}}(\alpha ,s,m=0)\,$ then its reciprocal is Weibull-distributed: $X^{-1}\sim {\textrm {Weibull}}(k=\alpha ,\lambda =s^{-1})\,$ 