gpd: The Generalized Pareto Distribution (GPD)

In eva: Extreme Value Analysis with Goodness-of-Fit Testing

The Generalized Pareto Distribution (GPD)

Description

Density, distribution function, quantile function and random number generation for the Generalized Pareto distribution with location, scale, and shape parameters.

Usage

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

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

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

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

Arguments

x	Vector of observations.
loc, scale, shape	Location, scale, and shape parameters. Can be vectors, but the lengths must be appropriate.
log.d	Logical; if TRUE, the log density is returned.
n	Number of observations.
p	Vector of probabilities.
lower.tail	Logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
log.p	Logical; if TRUE, probabilities p are given as log(p).
q	Vector of quantiles.

Details

The Generalized Pareto distribution function is given (Pickands, 1975) by

H(y) = 1 - \Big[1 + \frac{\xi (y - \mu)}{\sigma}\Big]^{-1/\xi}

defined on \{y : y > 0, (1 + \xi (y - \mu) / \sigma) > 0 \}, with location \mu, scale \sigma > 0, and shape parameter \xi.

References

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

Examples

dgpd(2:4, 1, 0.5, 0.01)
dgpd(2, -2:1, 0.5, 0.01)
pgpd(2:4, 1, 0.5, 0.01)
qgpd(seq(0.9, 0.6, -0.1), 2, 0.5, 0.01)
rgpd(6, 1, 0.5, 0.01)

## Generate sample with linear trend in location parameter
rgpd(6, 1:6, 0.5, 0.01)

## Generate sample with linear trend in location and scale parameter
rgpd(6, 1:6, seq(0.5, 3, 0.5), 0.01)

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

## Incorrect syntax (parameter vectors are of different lengths other than 1)
# rgpd(1, 1:8, 1:5, 0)

## Also incorrect syntax
# rgpd(10, 1:8, 1, 0.01)

eva documentation built on June 21, 2026, 9:07 a.m.