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gpd

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{ξ (y - μ)}{σ}\Big]^{-1/ξ}

defined on \{y : y > 0, (1 + ξ (y - μ) / σ) > 0 \}, with location μ, scale σ > 0, and shape parameter ξ.

References

Brian Bader, Jun Yan. "eva: Extreme Value Analysis with Goodness-of-Fit Testing." R package version (2016)

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)

tea

Threshold Estimation Approaches

v1.1

GPL-3

Authors

Johannes Ossberger

Initial release

2020-04-17

gpd