GenPareto: The Generalized Pareto Distribution (GPD)

In reservr: Fit Distributions and Neural Networks to Censored and Truncated Data

The Generalized Pareto Distribution (GPD)

Description

These functions provide information about the generalized Pareto distribution with threshold u. dgpd gives the density, pgpd gives the distribution function, qgpd gives the quantile function and rgpd generates random deviates.

Usage

rgpd(n = 1L, u = 0, sigmau = 1, xi = 0)

dgpd(x, u = 0, sigmau = 1, xi = 0, log = FALSE)

pgpd(q, u = 0, sigmau = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

qgpd(p, u = 0, sigmau = 1, xi = 0, lower.tail = TRUE, log.p = FALSE)

Arguments

n	integer number of observations.
u	threshold parameter (minimum value).
sigmau	scale parameter (must be positive).
xi	shape parameter
x, q	vector of quantiles.
log, log.p	logical; if TRUE, probabilities/densities p are given as log(p).
lower.tail	logical; if TRUE (default), probabilities are P(X \le x), otherwise P(X > x).
p	vector of probabilities.

Details

If u, sigmau or xi are not specified, they assume the default values of 0, 1 and 0 respectively.

The generalized Pareto distribution has density

f(x) = 1 / \sigma_u (1 + \xi z)^(- 1 / \xi - 1)

where z = (x - u) / \sigma_u and f(x) = exp(-z) if \xi is 0. The support is x \ge u for \xi \ge 0 and u \le x \le u - \sigma_u / \xi for \xi < 0.

The Expected value exists if \xi < 1 and is equal to

E(X) = u + \sigma_u / (1 - \xi)

k-th moments exist in general for k\xi < 1.

Value

rgpd generates random deviates.

dgpd gives the density.

pgpd gives the distribution function.

qgpd gives the quantile function.

References

https://en.wikipedia.org/wiki/Generalized_Pareto_distribution

Examples

x <- rgpd(1000, u = 1, sigmau = 0.5, xi = 0.1)
xx <- seq(-1, 10, 0.01)
hist(x, breaks = 100, freq = FALSE, xlim = c(-1, 10))
lines(xx, dgpd(xx, u = 1, sigmau = 0.5, xi = 0.1))

plot(xx, dgpd(xx, u = 1, sigmau = 1, xi = 0), type = "l")
lines(xx, dgpd(xx, u = 0.5, sigmau = 1, xi = -0.3), col = "blue", lwd = 2)
lines(xx, dgpd(xx, u = 1.5, sigmau = 1, xi = 0.3), col = "red", lwd = 2)

plot(xx, dgpd(xx, u = 1, sigmau = 1, xi = 0), type = "l")
lines(xx, dgpd(xx, u = 1, sigmau = 0.5, xi = 0), col = "blue", lwd = 2)
lines(xx, dgpd(xx, u = 1, sigmau = 2, xi = 0), col = "red", lwd = 2)

reservr documentation built on June 24, 2024, 5:10 p.m.