Fréchet distribution

Probability density function

Cumulative distribution function

Parameters shape.
(Optionally, two more parameters)
scale (default: )
location of minimum (default: )
Ex. kurtosis
Entropy , where is the Euler–Mascheroni constant.
MGF [1] Note: Moment exists if
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

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

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 has standardized moment

(with ) defined only for :

where is the Gamma function.

In particular:

The quantile of order can be expressed through the inverse of the distribution,


In particular the median is:

The mode of the distribution is

Especially for the 3-parameter Fréchet, the first quartile is and the third quartile

Also the quantiles for the mean and mode are:

Fitted cumulative Fréchet distribution to extreme one-day rainfalls


Related distributions


See also


  1. 1 2 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. Khan M.S.; Pasha G.R.; Pasha A.H. (February 2008). "Theoretical Analysis of Inverse Weibull Distribution" (PDF). WSEAS TRANSACTIONS on MATHEMATICS. 7 (2). pp. 30–38.
  3. de Gusmão, FelipeR.S. and Ortega, EdwinM.M. and Cordeiro, GaussM. (2011). "The generalized inverse Weibull distribution". Statistical Papers. 52 (3). Springer-Verlag. pp. 591–619. doi:10.1007/s00362-009-0271-3. ISSN 0932-5026.
  4. Coles, Stuart (2001). An Introduction to Statistical Modeling of Extreme Values,. Springer-Verlag. ISBN 1-85233-459-2.


External links

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