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Fréchet distribution - Wikipedia

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Fréchet

Probability density function

PDF of the Fréchet distribution

Cumulative distribution function

CDF of the Fréchet distribution
Parameters {\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\ })
Support {\displaystyle \ x>m\ }
PDF {\displaystyle \ {\frac {\ \alpha \ }{s}}\left({\frac {\ x-m\ }{s}}\right)^{-1-\alpha }~e^{-({\frac {x-m}{s}})^{-\alpha }}\ }
CDF {\displaystyle \ e^{-({\frac {x-m}{s}})^{-\alpha }}\ }
Quantile {\displaystyle \ \left[\ -\ln p\ \right]^{-{\tfrac {1}{\alpha }}}\ s+m\ }
Mean {\displaystyle {\begin{cases}\ m\ +\ s\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)~~&{\text{for }}~\alpha >1\\\ \infty &{\text{otherwise}}\end{cases}}}
Median {\displaystyle \ m\ +\ {\frac {s}{\ {\sqrt {\alpha \ }}\ {\log _{e}(2)}\ }}}
Mode {\displaystyle \ m\ +\ s\left({\frac {\alpha }{\ 1+\alpha }}\right)^{1/\alpha \ }}
Variance {\displaystyle {\begin{cases}\ s^{2}\left[\ \Gamma \left(1-{\tfrac {2}{\alpha }}\right)-\left[\Gamma \left(1-{\tfrac {1}{\alpha }}\right)\right]^{2}\ \right]~~&{\text{for }}~\alpha >2\\\ \infty &{\text{otherwise}}\end{cases}}}
Skewness {\displaystyle {\begin{cases}\ {\frac {\ A\ }{\ {\sqrt {B^{3}\ }}\ }}~~&{\text{for }}~\alpha >3\\\ \infty &{\text{otherwise}}\end{cases}}}
{\displaystyle {\begin{aligned}{\text{where}}~~A\ \equiv \ &\Gamma \left(1-{\tfrac {3}{\alpha }}\right)\\&-\ 3\ \Gamma \left(1-{\tfrac {2}{\alpha }}\right)\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)\\&\quad +\ 2{\Bigl [}\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)\ {\Bigr ]}^{3}\ \end{aligned}}}
{\displaystyle ~\quad {\text{and}}~~B\ \equiv \ \Gamma \left(1-{\tfrac {2}{\alpha }}\right)\ -\ {\Bigr [}\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)\ {\Bigr ]}^{2}~.}
Excess kurtosis {\displaystyle {\begin{cases}\ -6\ +\ {\frac {\ C\ }{\;D^{2}}}~~&{\text{for }}~\alpha >4\\\ \infty &{\text{otherwise}}\end{cases}}}
{\displaystyle {\begin{aligned}{\text{where}}~~C\ \equiv \ &\Gamma \left(1-{\tfrac {4}{\alpha }}\right)\\&-\ 4\ \Gamma \left(1-{\tfrac {3}{\alpha }}\right)\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)\\&\qquad +\ 3\ {\Bigl [}\ \Gamma \left(1-{\tfrac {2}{\alpha }}\right)\ {\Bigr ]}^{2}\ \end{aligned}}}
{\displaystyle ~\quad {\text{and}}~~D\ \equiv \ \Gamma \left(1-{\tfrac {2}{\alpha }}\right)\ -\ {\Bigl [}\ \Gamma \left(1-{\tfrac {1}{\alpha }}\right)\ {\Bigr ]}^{2}~.}
Entropy {\displaystyle \ 1+{\frac {\gamma }{\alpha }}+\gamma _{e}+\ln \left({\frac {s}{\alpha }}\right)\ ,} where {\displaystyle \ \gamma _{e}\ } is the Euler–Mascheroni constant.
MGF [1] Note: Moment {\displaystyle \ k\ } exists if {\displaystyle \ \alpha >k\ }
CF [1]

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~.}

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\ )=\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,[4] further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958.[5][6]

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

{\displaystyle \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 {\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:

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

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]

Fitted decline curve analysis. Duong model can be thought of as a generalization of the Frechet distribution.
Scaling relations
  1. ^ a b Muraleedharan, G.; Guedes Soares, C.; Lucas, Cláudia (2011). "Characteristic and moment generating functions of generalised extreme value distribution (GEV)". In Wright, Linda L. (ed.). Sea Level Rise, Coastal Engineering, Shorelines, and Tides. Nova Science Publishers. Chapter 14, pp. 269–276. 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): 30–38.
  3. ^ de Gusmão, Felipe R.S.; Ortega, Edwin M.M.; Cordeiro, Gauss M. (2011). "The generalized inverse Weibull distribution". Statistical Papers. 52 (3). Springer-Verlag: 591–619. doi:10.1007/s00362-009-0271-3. ISSN 0932-5026.
  4. ^ Fréchet, M. (1927). "Sur la loi de probabilité de l'écart maximum" [On the probability distribution of the maximum deviation]. Annales Polonici Mathematici (in French). 6: 93.
  5. ^ Fisher, R.A.; Tippett, L.H.C. (1928). "Limiting forms of the frequency distribution of the largest and smallest member of a sample". Proceedings of the Cambridge Philosophical Society. 24 (2): 180–190. Bibcode:1928PCPS...24..180F. doi:10.1017/S0305004100015681. S2CID 123125823.
  6. ^ Gumbel, E.J. (1958). Statistics of Extremes. New York, NY: Columbia University Press. OCLC 180577.
  7. ^ Coles, Stuart (2001). An Introduction to Statistical Modeling of Extreme Values. Springer-Verlag. ISBN 978-1-85233-459-8.
  8. ^ Lee, Se Yoon; Mallick, Bani (2021). "Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas". Sankhya B. 84: 1–43. doi:10.1007/s13571-020-00245-8.
  • Kotz, S.; Nadarajah, S. (2000). Extreme Value Distributions: Theory and applications. World Scientific. ISBN 1-86094-224-5.