Fréchet distribution

From Wikipedia, the free encyclopedia

Fréchet
Probability density function
Cumulative distribution function
Parameters \alpha \in (0,\infty] shape
Support x > 0
Probability density function (pdf) \alpha \; x^{-1-\alpha} \; e^{-x^{-\alpha}}
Cumulative distribution function (cdf) e^{-x^{-\alpha}}
Mean \Gamma\left(1-\frac{1}{\alpha}\right) \text{ if } \alpha>1
Median \left(\frac{1}{\log_e(2)}\right)^{1/\alpha}
Mode \left(\frac{\alpha}{1+\alpha}\right)^{1/\alpha}
Variance \Gamma\left(1-\frac{2}{\alpha}\right)- \left(\Gamma\left(1-\frac{1}{\alpha}\right)\right)^2\text{ if } \alpha>2
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

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

Pr(X<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 and a scale parameter s>0 with the cumulative probability function

Pr(X<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

[edit] See also

[edit] External links

[edit] Publications

  • 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.


This probability-related article is a stub. You can help Wikipedia by expanding it.
v • d • e
Probability distributions
 
Continuous univariate supported on a semi-infinite interval
Languages