The generalised Pareto distribution
Density, distribution function, quantile function and random generation for the Generalised Pareto Distribution (GPD).
dgpd(x, gamma, mu = 0, sigma, log = FALSE) pgpd(x, gamma, mu = 0, sigma, lower.tail = TRUE, log.p = FALSE) qgpd(p, gamma, mu = 0, sigma, lower.tail = TRUE, log.p = FALSE) rgpd(n, gamma, mu = 0, sigma)
x |
Vector of quantiles. |
p |
Vector of probabilities. |
n |
Number of observations. |
gamma |
The γ parameter of the GPD, a real number. |
mu |
The μ parameter of the GPD, a strictly positive number. Default is 0. |
sigma |
The σ parameter of the GPD, a strictly positive number. |
log |
Logical indicating if the densities are given as \log(f), default is |
lower.tail |
Logical indicating if the probabilities are of the form P(X≤ x) ( |
log.p |
Logical indicating if the probabilities are given as \log(p), default is |
The Cumulative Distribution Function (CDF) of the GPD for γ \neq 0 is equal to F(x) = 1-(1+γ (x-μ)/σ)^{-1/γ} for all x ≥ μ and F(x)=0 otherwise. When γ=0, the CDF is given by F(x) = 1-\exp((x-μ)/σ) for all x ≥ μ and F(x)=0 otherwise.
dgpd gives the density function evaluated in x, pgpd the CDF evaluated in x and qgpd the quantile function evaluated in p. The length of the result is equal to the length of x or p.
rgpd returns a random sample of length n.
Tom Reynkens.
Beirlant J., Goegebeur Y., Segers, J. and Teugels, J. (2004). Statistics of Extremes: Theory and Applications, Wiley Series in Probability, Wiley, Chichester.
# Plot of the PDF x <- seq(0, 10, 0.01) plot(x, dgpd(x, gamma=1/2, sigma=5), xlab="x", ylab="PDF", type="l") # Plot of the CDF x <- seq(0, 10, 0.01) plot(x, pgpd(x, gamma=1/2, sigma=5), xlab="x", ylab="CDF", type="l")
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