Fréchet distribution

Characteristics

• For the expectation is
• For the variance is

Applications

• 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.

• 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 distributions

• If 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:

  Properties

• The Frechet distribution is a max stable distribution
• The negative of a random variable having a Frechet distribution is a min stable distribution

  Publications

• Fréchet, M.,. "Sur la loi de probabilité de l'écart maximum." Ann. Soc. Polon. Math. 6, 93.
• Fisher, R.A., Tippett, L.H.C.,. "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.. "Statistics of Extremes." Columbia University Press, New York.
• Kotz, S.; Nadarajah, S. Extreme value distributions: theory and applications, World Scientific.