See https://en.wikipedia.org/wiki/Generalized_extreme_value_distribution for an introduction to the GEV distribution.
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 \(k<0\).
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
$$\alpha = \frac{\lambda_2 k}{(1-2^{-k})\Gamma(1+k)}$$
$$\xi = \lambda_1 - \alpha[1 - \Gamma(1+k)]/k$$
Lmom.GEV and par.GEV accept input as vectors of equal length. In f.GEV, F.GEV, invF.GEV and rand.GEV parameters (xi, alfa, k) must be atomic.