Fréchet distribution

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 is a shape parameter. It can be generalised to include a location parameter and a scale parameter 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.

Characteristics

The single parameter Fréchet, with parameter has standardized moment
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:

Properties

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

Related distributions

The cumulative distribution function of the Frechet distribution solves the maximum stability postulate equation.
Scaling relations include:

If then
If then its reciprocal is Weibull-distributed:
If then
If and then

Applications

In hydrology, the Fréchet distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges. This picture 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.
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. 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.
In economics it is used to model the idiosyncratic component of preferences of individuals for different products, locations, or firms.