GEV provides the link between L-moments of a sample and the three parameter
generalized extreme value distribution.f.GEV (x, xi, alfa, k)
F.GEV (x, xi, alfa, k)
invF.GEV (F, xi, alfa, k)
Lmom.GEV (xi, alfa, k)
par.GEV (lambda1, lambda2, tau3)
rand.GEV (numerosita, xi, alfa, k)f.GEV gives the density $f$, F.GEV gives the distribution function $F$, invF.GEV gives
the quantile function $x$, Lmom.GEV gives the L-moments ($\lambda_1$, $\lambda_2$, $\tau_3$, $\tau_4$), par.GEV gives the parameters (xi, alfa, k), and rand.GEV generates random deviates.Definition
Parameters (3): $\xi$ (location), $\alpha$ (scale), $k$ (shape).
Range of $x$: $-\infty < x \le \xi + \alpha / k$ if $k>0$; $-\infty < x < \infty$ if $k=0$; $\xi + \alpha / k \le x < \infty$ if $k<0$.< p="">
Probability density function: $$f(x) = \alpha^{-1} e^{-(1-k)y - e^{-y}}$$ where $y = -k^{-1}\log{1 - k(x - \xi)/\alpha}$ if $k \ne 0$, $y = (x-\xi)/\alpha$ if $k=0$.
Cumulative distribution function: $$F(x) = e^{-e^{-y}}$$
Quantile function: $x(F) = \xi + \alpha[1-(-\log F)^k]/k$ if $k \ne 0$, $x(F) = \xi - \alpha \log(-\log F)$ if $k=0$.
$k=0$ is the Gumbel distribution; $k=1$ is the reverse exponential distribution.
L-moments
L-moments are defined for $k>-1$.
$$\lambda_1 = \xi + \alpha[1 - \Gamma (1+k)]/k$$ $$\lambda_2 = \alpha (1-2^{-k}) \Gamma (1+k)]/k$$ $$\tau_3 = 2(1-3^{-k})/(1-2^{-k})-3$$ $$\tau_4 = [5(1-4^{-k})-10(1-3^{-k})+6(1-2^{-k})]/(1-2^{-k})$$
Here $\Gamma$ denote the gamma function $$\Gamma (x) = \int_0^{\infty} t^{x-1} e^{-t} dt$$
Parameters
To estimate $k$, no explicit solution is possible, but the following approximation has accurancy better than $9 \times 10^{-4}$ for $-0.5 \le \tau_3 \le 0.5$: $$k \approx 7.8590 c + 2.9554 c^2$$ where $$c = \frac{2}{3+\tau_3} - \frac{\log 2}{\log 3}$$ The other parameters are then given by $$\alpha = \frac{\lambda_2 k}{(1-2^{-k})\Gamma(1+k)}$$ $$\xi = \lambda_1 - \alpha[1 - \Gamma(1+k)]/k$$
rnorm, runif, KAPPA, Lmoments.