Generalized Extreme Value Distribution

Three-parameter Generalised Extreme Value (GEV) distribution with location $\mu$, scale $\sigma$, and shape $\xi$. The GEV encompasses the Gumbel ($\xi = 0$), Frechet ($\xi > 0$), and reversed Weibull ($\xi < 0$) families. Density, distribution, quantile, and random generation functions are implemented natively. Fitting is performed via closed-form L-moments through pelgev.

Define $t(x) = (1 + \xi(x-\mu)/\sigma)^{-1/\xi}$. The probability density function is: $$f(x) = \frac{1}{\sigma} t(x)^{\xi+1} \exp(-t(x)), \quad \xi \neq 0$$ $$f(x) = \frac{1}{\sigma} \exp\left(-\frac{x-\mu}{\sigma} - \exp\left(-\frac{x-\mu}{\sigma}\right)\right), \quad \xi = 0$$ with $1 + \xi(x-\mu)/\sigma > 0$. where:

$\mu$ — location parameter (location)
$\sigma$ — scale parameter (scale)
$\xi$ — shape parameter (shape)

Usage

dgev(x, location, scale, shape)

pgev(q, location, scale, shape)

qgev(p, shape, scale, location)

rgev(n, location, scale, shape)

Arguments

x, q: vector of quantiles.
location: location parameter.
scale: scale parameter.
shape: shape parameter.
p: vector of probabilities.
n: number of observations. If length(n) > 1, the length is taken as the number required.