Generalized Pareto Distribution — GenPareto • anyFit

Three-parameter Generalised Pareto distribution (GPD) with location $\mu$, scale $\sigma$, and shape $\xi$. The GPD is the limiting distribution of peaks over a threshold and is widely used in extreme-value modelling. The distribution is lower-bounded at the location parameter. Density, distribution, quantile, and random generation functions are provided via the lmom package. Fitting is performed via closed-form L-moments through pelgpa; an optional fixed lower bound is supported.

Define $z = (x - \mu)/\sigma$. The probability density function is: $$f(x) = \frac{1}{\sigma} (1 - \xi z)^{1/\xi - 1}, \quad \xi \neq 0$$ $$f(x) = \frac{1}{\sigma} e^{-z}, \quad \xi = 0$$ with $z \ge 0$ and $1 - \xi z > 0$. where:

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

Usage

dgpd(x, location, scale, shape)

pgpd(q, location, scale, shape)

qgpd(p, location, scale, shape)

rgpd(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.