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Concept
Fréchet distribution
Summary
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
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:
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:
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).
In decline curve analysis, a declining pattern the time series data of oil or gas production rate over time for a well can be described by the Fréchet 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 .
Official source
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