Generalized extreme value distribution with location 0, scale 1 and shape
xi, \(F(z) = \exp\{-(1 + \xi z)^{-1/\xi}\}\) on \(1 + \xi z > 0\)
(Gumbel when \(\xi = 0\)). The shape is restricted to (-0.5, 0.5):
above -0.5 for regular likelihood properties, below 0.5 for a finite
variance, which the GARCH mode power \(\delta = 2\) requires.
Details
The innovations are not centered: \(E[z] = (\Gamma(1-\xi) - 1)/\xi\), so
the model parameter mu is a location, not the conditional mean.
Examples
d <- gs_gev()
d$mean(c(xi = 0.1))
#> [1] 0.686287