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
Probability density function
Cumulative distribution function
(Optionally, two more parameters)
scale (default: )
location of minimum (default: )
|Entropy||, where is the Euler–Mascheroni constant.|
|MGF||Note: Moment exists if|
The single parameter Fréchet with parameter has standardized moment
(with ) defined only for :
where is the Gamma function.
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:
- 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).
- 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 and then mapping from Cartesian to pseudo-polar coordinates . Values of correspond to the extreme data for which at least one component is large while approximately 1 or 0 corresponds to only one component being extreme.
- 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
- 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.
- 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.CS1 maint: uses authors parameter (link)
- Coles, Stuart (2001). An Introduction to Statistical Modeling of Extreme Values. Springer-Verlag. ISBN 978-1-85233-459-8.
- 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