Generalized Pareto distribution
Density, distribution function, quantile function and random generation for the generalized Pareto distribution.
dgpd(x, mu = 0, sigma = 1, xi = 0, log = FALSE) pgpd(q, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE) qgpd(p, mu = 0, sigma = 1, xi = 0, lower.tail = TRUE, log.p = FALSE) rgpd(n, mu = 0, sigma = 1, xi = 0)
x, q |
vector of quantiles. |
mu, sigma, xi |
location, scale, and shape parameters. Scale must be positive. |
log, log.p |
logical; if TRUE, probabilities p are given as log(p). |
lower.tail |
logical; if TRUE (default), probabilities are P[X ≤ x] otherwise, P[X > x]. |
p |
vector of probabilities. |
n |
number of observations. If |
Probability density function
f(x) = [if ξ != 0:] (1+ξ*(x-μ)/σ)^{-(ξ+1)/ξ}/σ [else:] exp(-(x-μ)/σ)/σ
Cumulative distribution function
F(x) = [if ξ != 0:] 1-(1+ξ*(x-μ)/σ)^{-1/ξ} [else:] 1-exp(-(x-μ)/σ)
Quantile function
F^-1(x) = [if ξ != 0:] μ + σ * ((1-p)^{-ξ}-1)/ξ [else:] μ - σ * log(1-p)
Coles, S. (2001). An Introduction to Statistical Modeling of Extreme Values. Springer.
x <- rgpd(1e5, 5, 2, .1) hist(x, 100, freq = FALSE, xlim = c(0, 50)) curve(dgpd(x, 5, 2, .1), 0, 50, col = "red", add = TRUE, n = 5000) hist(pgpd(x, 5, 2, .1)) plot(ecdf(x)) curve(pgpd(x, 5, 2, .1), 0, 50, col = "red", lwd = 2, add = TRUE)
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