GPD: The Generalized Pareto Distribution

In mistr: Mixture and Composite Distributions

The Generalized Pareto Distribution

Description

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

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	location parameter.
scale	scale parameter.
shape	shape parameter.
log, log.p	logical; if TRUE, probabilities p are given as log(p), default: FALSE.
lower.tail	logical; if TRUE, probabilities are P[X ≤ x] otherwise, P[X > x], default: TRUE.
p	vector of probabilities.
n	number of observations.

Details

The generalized Pareto distribution function with location parameter μ, scale parameter σ and shape parameter ξ has density given by

f(x)=1/σ (1 + ξ z)^-(1/ξ + 1)

for x≥ μ and ξ> 0, or μ-σ/ξ ≥ x≥ μ and ξ< 0, where z=(x-μ)/σ. In the case where ξ= 0, the density is equal to f(x)=1/σ e^-z for x≥ μ. The cumulative distribution function is

F(x)=1-(1+ξ z)^(-1/ξ)

for x≥ μ and ξ> 0, or μ-σ/ξ ≥ x≥ μ and ξ< 0, with z as stated above. If ξ= 0 the CDF has form F(x)=1-e^-z.

See https://en.wikipedia.org/wiki/Generalized_Pareto_distribution for more details.

Value

dGPD gives the density, pGPD gives the distribution function, qGPD gives the quantile function, and rGPD generates random deviates.

Invalid arguments will result in return value NaN, with a warning.

See Also

GPDdist

Examples

dGPD(seq(1, 5), 0, 1, 1)
qGPD(pGPD(seq(1, 5), 0, 1, 1), 0, 1 ,1)
rGPD(5, 0, 1, 1)

mistr documentation built on March 7, 2023, 7:42 p.m.