Fréchet distribution

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Probability density function
PDF of the Fréchet distribution
Cumulative distribution function
CDF of the Fréchet distribution
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}}
Mean \begin{cases}
                  \ m+s\Gamma\left(1-\frac{1}{\alpha}\right)  & \text{for } \alpha>1  \\
                  \ \infty              & \text{otherwise}
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}
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}
Ex. 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}
Entropy  1 + \frac{\gamma}{\alpha} + \gamma +\ln \left( \frac{s}{\alpha} \right) , where \gamma is the Euler–Mascheroni constant.
MGF [1] Note: Moment k exists if \alpha>k
CF [1]

The Fréchet distribution is a special case of the generalized extreme value distribution. It has the cumulative distribution function

\Pr(X \le 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 \le 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.


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:


where \Gamma\left(z\right) is the Gamma function.

In particular:

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).
Fitted cumulative Fréchet distribution to extreme one-day rainfalls


Related distributions[edit]


See also[edit]


  1. ^ a b 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. ^ Coles, Stuart (2001). An Introduction to Statistical Modeling of Extreme Values,. Springer-Verlag. ISBN 1-85233-459-2. 


  • 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 24:180–190.
  • 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

External links[edit]