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Generalized Pareto Distribution

Source: R/dpqr-gpd.R
dpqr-gpd.Rd

The Generalized Pareto Distribution (GPD) is a two-parameter family of distributions used to model exceedances over a high threshold, commonly applied in extreme value theory as the limiting distribution of threshold excesses.

Usage

dgpd(x, loc = 0, scale = 1, shape = 0, log = FALSE)

pgpd(q, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)

qgpd(p, loc = 0, scale = 1, shape = 0, lower.tail = TRUE, log.p = FALSE)

rgpd(n, loc = 0, scale = 1, shape = 0)

Arguments

x, q

vector of quantiles.

loc, scale, shape

location, scale and shape parameters; the shape argument cannot be a vector (must have length one).

log, log.p

logical; if TRUE, probabilities p are given as log(p) and the density is returned on the log scale.

lower.tail

logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

p

vector of probabilities.

n

number of observations.

Value

dgpd() gives the density function, pgpd() gives the distribution function, qgpd() gives the quantile function, and rgpd() generates random deviates.

Details

Density function, distribution function, quantile function and random generation for the generalized Pareto distribution (GPD) with location, scale and shape parameters.

The generalized Pareto distribution function (Pickands, 1975) with parameters \(`loc` = a\), \(`scale` = b\) and \(`shape` = s\) is $$G(z) = 1 - \{1+s(z-a)/b\}^{-1/s}$$ for \(1+s(z-a)/b > 0\) and \(z > a\), where \(b > 0\). If \(s = 0\) the distribution is defined by continuity.

Note

Based on code by Alec Stephenson previously published in the evd package, adapted to conform to package standards.

References

Pickands, J. (1975) Statistical inference using extreme order statistics. Annals of Statistics, 3, 119–131.

See also

distributions-overview; evd::fpot() for fitting peaks-over-threshold models

Examples


dgpd(2:4, 1, 0.5, 0.8)
#> [1] 0.23299144 0.07919889 0.03831043
pgpd(2:4, 1, 0.5, 0.8)
#> [1] 0.6971111 0.8336823 0.8888998
qgpd(seq(0.9, 0.6, -0.1), 2, 0.5, 0.8)
#> [1] 5.318483 3.639936 3.012506 2.675864
rgpd(6, 1, 0.5, 0.8)
#> [1] 1.073178 1.729300 1.188356 2.335112 1.259598 1.012437
p <- (1:9)/10
pgpd(qgpd(p, 1, 2, 0.8), 1, 2, 0.8)
#> [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
## [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9