Generalized Extreme Value distribution (GEV)

$gev(x) = exp(-(1+\frac{\xi(x-\mu)}{\sigma})^{\frac{-1}{\xi}})$

This distribution can be derived using the block maxima method: all observations x_i are divided in equally spaced blocks. Then, we note the local maxima in each block. After normalization these maxima follow the GEV.

$\xi = 0$ : Gumbel distribution

$\xi < 0$ : Weibull distribution

$\xi > 0$ : Fréchet distribution

The Fréchet distribution shows significant fat tails.