Table 2 Marginal distributions

Extreme value (Kotz and Nadarajah 2000)

\(f(x;\mu , \sigma )= \sigma ^{-1}\exp (\frac{x-\mu }{\sigma })\exp \left( -\exp \left( \frac{x-\mu }{\sigma }\right) \right)\)

Location \(\mu\)  scale \(\sigma\) 

Generalized extreme value (Hosking et al. 1985)

\({\displaystyle f(x;\mu , \sigma ,\xi )={{\left\{ \begin{array}{ll}{\big (}1+\xi ({\tfrac{x-\mu }{\sigma }}){\big )}^{-1/\xi }&{}{\text {if}}\ \xi \ne 0\\ \mathrm{e}^{-(x-\mu )/\sigma }&{}{\text {if}}\ \xi =0\end{array}\right. }}}\)

Location \(\mu\)  scale \(\sigma\)  shape \(\xi\)

Generalized Pareto (Hosking and Wallis 1987)

\(f(x;\sigma ,\xi )= f_{{(\xi ,\mu ,\sigma )}}(x)={\frac{1}{\sigma }}\left( 1+{\frac{\xi (x-\mu )}{\sigma }}\right) ^{{\left( -{\frac{1}{\xi }}-1\right) }}\)

Location \(\mu\)  scale \(\sigma\)  shape \(\xi\)