Fréchet distribution

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

Fréchet

Probability density function
Cumulative distribution function
Parameters	$\alpha \in (0,\infty )$ shape. (Optionally, two more parameters) $s\in (0,\infty )$ scale (default: $s=1\,$) $m\in (-\infty ,\infty )$ location of minimum (default: $m=0\,$)
Support	$x>m$
PDF	${\frac {\alpha }{s}}\;\left({\frac {x-m}{s}}\right)^{-1-\alpha }\;e^{-({\frac {x-m}{s}})^{-\alpha }}$
CDF	$e^{-({\frac {x-m}{s}})^{-\alpha }}$
Quantile	$\left(-\ln(p)\right)^{-{\frac {1}{\alpha }}}$
Mean	${\begin{cases}\ m+s\Gamma \left(1-{\frac {1}{\alpha }}\right)&{\text{for }}\alpha >1\\\ \infty &{\text{otherwise}}\end{cases}}$
Median	$m+{\frac {s}{\sqrt[{\alpha }]{\log _{e}(2)}}}$
Mode	$m+s\left({\frac {\alpha }{1+\alpha }}\right)^{1/\alpha }$
Variance	${\begin{cases}\ s^{2}\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\left(\Gamma \left(1-{\frac {1}{\alpha }}\right)\right)^{2}\right)&{\text{for }}\alpha >2\\\ \infty &{\text{otherwise}}\end{cases}}$
Skewness	${\begin{cases}\ {\frac {\Gamma \left(1-{\frac {3}{\alpha }}\right)-3\Gamma \left(1-{\frac {2}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+2\Gamma ^{3}\left(1-{\frac {1}{\alpha }}\right)}{\sqrt {\left(\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right)^{3}}}}&{\text{for }}\alpha >3\\\ \infty &{\text{otherwise}}\end{cases}}$
Excess kurtosis	${\begin{cases}\ -6+{\frac {\Gamma \left(1-{\frac {4}{\alpha }}\right)-4\Gamma \left(1-{\frac {3}{\alpha }}\right)\Gamma \left(1-{\frac {1}{\alpha }}\right)+3\Gamma ^{2}\left(1-{\frac {2}{\alpha }}\right)}{\left[\Gamma \left(1-{\frac {2}{\alpha }}\right)-\Gamma ^{2}\left(1-{\frac {1}{\alpha }}\right)\right]^{2}}}&{\text{for }}\alpha >4\\\ \infty &{\text{otherwise}}\end{cases}}$
Entropy	$1+{\frac {\gamma }{\alpha }}+\gamma +\ln \left({\frac {s}{\alpha }}\right)$, where $\gamma$ is the Euler–Mascheroni constant.
MGF	Note: Moment $k$ exists if $\alpha >k$
CF

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

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

$\Pr(\ X\leq x\ )=\exp \left(\ -\left({\tfrac {\ x-m\ }{s}}\right)^{-\alpha }\ \right)~{\text{ if }}~x>m~.$

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

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

(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

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. 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).

Fitted decline curve analysis. Duong model can be thought of as a generalization of the Frechet distribution.

• In decline curve analysis, a declining pattern the time series data of oil or gas production rate over time for a well can be described by the Fréchet 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 $Z_{i}=-1/\log F_{i}(X_{i})$ 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.
• In Economics it is used to model the idiosyncratic component of preferences of individuals for different products (Industrial Organization), locations (Urban Economics), or firms (Labor Economics).

Related distributions

• The cumulative distribution function of the Frechet distribution solves the maximum stability postulate equation

Scaling relations

• If $\ X\sim U(\ 0,1\ )\$ (continuous uniform distribution) then $\ m+s\cdot {\Bigl (}-\log _{e}\!(X)\ {\Bigr )}^{\frac {-1\;}{\alpha }}\sim {\textsf {Frechet}}(\alpha ,s,m)\$

• If $\ X\sim {\textsf {Frechet}}(\ \alpha ,s,m=0\ )\$ then its reciprocal is Weibull-distributed: $\ {\frac {\ 1\ }{X}}\sim {\textsf {Weibull}}\!\left(\ k=\alpha ,\lambda ={\tfrac {1}{s}}\ \right)\$

• If $\ X\sim {\textsf {Frechet}}(\alpha ,s,m)\$ then $\ k\ X+b\sim {\textrm {Frechet}}(\ \alpha ,ks,k\ m+b\ )\$

• If $\ X_{i}\sim {\textsf {Frechet}}(\ \alpha ,s,m\ )\$ and $\ Y=\max\{\ X_{1},\ldots ,X_{n}\ \}\$ then $\ Y\sim {\textsf {Frechet}}(\ \alpha ,n^{\tfrac {1}{\alpha }}s,m\ )\$

Properties

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

See also

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

Further reading

• Kotz, S.; Nadarajah, S. (2000). Extreme Value Distributions: Theory and applications. World Scientific. ISBN 1-86094-224-5.

External links

• Hurairah, Ahmed; Ibrahim, Noor Akma; bin Daud, Isa; Haron, Kassim (February 2005). "An application of a new extreme value distribution to air pollution data". Management of Environmental Quality. 16 (1): 17–25. doi:10.1108/14777830510574317. ISSN 1477-7835.
• "wfrechstat: Mean and variance for the Frechet distribution". Wave Analysis for Fatigue and Oceanography (WAFO) (Matlab software & docs). Centre for Mathematical Science. Lund University / Lund Institute of Technology. Retrieved 11 November 2023 – via www.maths.lth.se.