Fréchet distribution

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:

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

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

Fitted cumulative Fréchet distribution to extreme one-day rainfalls

• In hydrology, the Fréchet distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges.[7] The blue picture, made with CumFreq, illustrates an example of fitting the Fréchet distribution to ranked annually maximum one-day rainfalls in Oman showing also the 90% confidence belt based on the binomial distribution. The cumulative frequencies of the rainfall data are represented by plotting positions as part of the cumulative frequency analysis.

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: ${\displaystyle X^{-1}\sim {\textrm {Weibull}}(k=\alpha ,\lambda =s^{-1})\,}$

• The Frechet distribution is a max stable distribution
• The negative of a random variable having a Frechet distribution is a min stable distribution

• Type-2 Gumbel distribution
• Fisher–Tippett–Gnedenko theorem

• Kotz, S.; Nadarajah, S. (2000) Extreme value distributions: theory and applications, World Scientific. ISBN 1-86094-224-5

• An application of a new extreme value distribution to air pollution data
• Wave Analysis for Fatigue and Oceanography