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.

Characteristics

The single parameter Fréchet, with parameter \ \alpha, has standardized moment (with ) defined only for where is the Gamma function. In particular: The quantile q_y of order y 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:

Applications

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).

Related distributions

Properties