Fréchet distribution

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

Fréchet
Probability density function
PDF of the Fréchet distribution
Cumulative distribution function
CDF of the Fréchet distribution
Parameters shape.
(Optionally, two more parameters)
scale (default: )
location of minimum (default: )
Support
PDF
CDF
Mean
Median
Mode
Variance
Skewness
Ex. kurtosis
Entropy , where is the Euler–Mascheroni constant.
MGF [1] Note: Moment exists if
CF [1]

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

CharacteristicsEdit

The single parameter Fréchet with parameter   has standardized moment

 

(with  ) defined only for  :

 

where   is the Gamma function.

In particular:

  • For   the expectation is  
  • For   the variance is  

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:

 
 

ApplicationsEdit

 
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]

  • 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.

Related distributionsEdit

  • If   (Uniform distribution (continuous)) then  
  • If   then  
  • If   and   then  
  • The cumulative distribution function of the Frechet distribution solves the maximum stability postulate equation
  • If   then its reciprocal is Weibull-distributed:  

PropertiesEdit

See alsoEdit

ReferencesEdit

  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. ^ 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.CS1 maint: uses authors parameter (link)
  4. ^ Fréchet, M. (1927). "Sur la loi de probabilité de l'écart maximum". Ann. Soc. Polon. Math. 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". Proc. Cambridge Philosophical Society. 24 (2): 180–190. doi:10.1017/S0305004100015681.
  6. ^ Gumbel, E. J. (1958). Statistics of Extremes. New York: 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.

Further readingEdit

  • Kotz, S.; Nadarajah, S. (2000) Extreme value distributions: theory and applications, World Scientific. ISBN 1-86094-224-5

External linksEdit