The Extended Pareto Distribution
Density, distribution function, quantile function and random generation for the Extended Pareto Distribution (EPD).
depd(x, gamma, kappa, tau = -1, log = FALSE) pepd(x, gamma, kappa, tau = -1, lower.tail = TRUE, log.p = FALSE) qepd(p, gamma, kappa, tau = -1, lower.tail = TRUE, log.p = FALSE) repd(n, gamma, kappa, tau = -1)
x |
Vector of quantiles. |
p |
Vector of probabilities. |
n |
Number of observations. |
gamma |
The γ parameter of the EPD, a strictly positive number. |
kappa |
The κ parameter of the EPD. It should be larger than \max\{-1,1/τ\}. |
tau |
The τ parameter of the EPD, a strictly negative number. Default is -1. |
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 EPD is equal to F(x) = 1-(x(1+κ-κ x^{τ}))^{-1/γ} for all x > 1 and F(x)=0 otherwise.
Note that an EPD random variable with τ=-1 and κ=γ/σ-1 is GPD distributed with μ=1, γ and σ.
depd gives the density function evaluated in x, pepd the CDF evaluated in x and qepd the quantile function evaluated in p. The length of the result is equal to the length of x or p.
repd returns a random sample of length n.
Tom Reynkens.
Beirlant, J., Joossens, E. and Segers, J. (2009). "Second-Order Refined Peaks-Over-Threshold Modelling for Heavy-Tailed Distributions." Journal of Statistical Planning and Inference, 139, 2800–2815.
# Plot of the PDF x <- seq(0, 10, 0.01) plot(x, depd(x, gamma=1/2, kappa=1, tau=-1), xlab="x", ylab="PDF", type="l") # Plot of the CDF x <- seq(0, 10, 0.01) plot(x, pepd(x, gamma=1/2, kappa=1, tau=-1), xlab="x", ylab="CDF", type="l")
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