Extended Pareto {ReIns}

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.

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.

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.

Author(s)

Tom Reynkens.

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

# 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")