The generalized extreme-value distribution with
location parameter \(\xi\),
scale parameter \(\alpha\) and
shape parameter \(k\) has distribution function
$$F(x)=\exp\lbrace-\exp(-y)\rbrace$$ where
$$y=-k^{-1}\log\lbrace1-k(x-\xi)/\alpha\rbrace,$$
with \(x\) bounded by \(\xi+\alpha/k\)
from below if \(k<0\) and from above if \(k>0\),
and quantile function
$$x(F)=\xi+{\alpha\over k}\lbrace1-(-\log F)^k\rbrace.$$
Extreme-value distribution types I, II and III (Gumbel, Frechet, Weibull)
correspond to shape parameter values
\(k=0\), \(k<0\) and \(k>0\) respectively.