ReIns (version 1.0.7)

Extended Pareto: The Extended Pareto Distribution

Description

Density, distribution function, quantile function and random generation for the Extended Pareto Distribution (EPD).

Usage

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)

Arguments

x

Vector of quantiles.

p

Vector of probabilities.

n

Number of observations.

gamma

The \(\gamma\) parameter of the EPD, a strictly positive number.

kappa

The \(\kappa\) parameter of the EPD. It should be larger than \(\max\{-1,1/\tau\}\).

tau

The \(\tau\) parameter of the EPD, a strictly negative number. Default is -1.

log

Logical indicating if the densities are given as \(\log(f)\), default is FALSE.

lower.tail

Logical indicating if the probabilities are of the form \(P(X\le x)\) (TRUE) or \(P(X>x)\) (FALSE). Default is TRUE.

log.p

Logical indicating if the probabilities are given as \(\log(p)\), default is FALSE.

Value

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 \(n\).

Details

The Cumulative Distribution Function (CDF) of the EPD is equal to \(F(x) = 1-(x(1+\kappa-\kappa x^{\tau}))^{-1/\gamma}\) for all \(x > 1\) and \(F(x)=0\) otherwise.

Note that an EPD random variable with \(\tau=-1\) and \(\kappa=\gamma/\sigma-1\) is GPD distributed with \(\mu=1\), \(\gamma\) and \(\sigma\).

References

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.

See Also

Pareto, GPD, Distributions

Examples

# NOT RUN {
# 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")
# }