GUMBEL provides the link between L-moments of a sample and the two parameter
Gumbel distribution.f.gumb (x, xi, alfa)
F.gumb (x, xi, alfa)
invF.gumb (F, xi, alfa)
Lmom.gumb (xi, alfa)
par.gumb (lambda1, lambda2)
rand.gumb (numerosita, xi, alfa)f.gumb gives the density $f$, F.gumb gives the distribution function $F$, invF.gumb gives
the quantile function $x$, Lmom.gumb gives the L-moments ($\lambda_1$, $\lambda_2$, $\tau_3$, $\tau_4$)), par.gumb gives the parameters (xi, alfa), and rand.gumb generates random deviates.Parameters (2): $\xi$ (location), $\alpha$ (scale).
Range of $x$: $-\infty < x < \infty$.
Probability density function: $$f(x) = \alpha^{-1} \exp[-(x-\xi)/\alpha] \exp{- \exp[-(x-\xi)/\alpha]}$$
Cumulative distribution function: $$F(x) = \exp[-\exp(-(x-\xi)/\alpha)]$$
Quantile function: $x(F) = \xi - \alpha \log(-\log F)$.
L-moments
$$\lambda_1 = \xi + \alpha \gamma$$ $$\lambda_2 = \alpha \log 2$$ $$\tau_3 = 0.1699 = \log(9/8)/ \log 2$$ $$\tau_4 = 0.1504 = (16 \log 2 - 10 \log 3)/ \log 2$$
Here $\gamma$ is Euler's constant, 0.5772...
Parameters
$$\alpha=\lambda_2 / \log 2$$ $$\xi = \lambda_1 - \gamma \alpha$$
rnorm, runif, GEV, Lmoments.