RTDE (version 0.2-2)

EPD: The Extended Pareto Distribution

Description

Density function, distribution function, quantile function, random generation.

Usage

dEPD(x, eta, delta, rho, tau, log = FALSE)
pEPD(q, eta, delta, rho, tau, lower.tail=TRUE, log.p = FALSE)
qEPD(p, eta, delta, rho, tau, lower.tail=TRUE, log.p = FALSE,
    control=list())
rEPD(n, eta, delta, rho, tau)

Value

dEPD gives the density,

pEPD gives the distribution function,

qEPD gives the quantile function, and

rEPD generates random deviates.

The length of the result is determined by n for

rEPD, and is the maximum of the lengths of the numerical parameters for the other functions.

The numerical parameters other than n are recycled to the length of the result. Only the first elements of the logical parameters are used.

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

eta

first shape parameter.

delta

nuisance parameter.

rho, tau

second shape parameter.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).

control

A list of control paremeters. See section Details.

Author

Christophe Dutang

Details

The extended Pareto distribution is defined by the following density $$ f(x) = \frac{1}{\eta} x^{-1/\eta-1}[1+\delta(1-x^{-\tau})]^{-1/\eta-1}[1+\delta(1-(1-\tau)x^{-\tau})] $$ for all \(x>1\) when parametrized by \(\tau\). However, a typical parametrization is obtained by setting \(\tau=-\rho/\eta\), i.e. $$ f(x) = \frac{1}{\eta} x^{-1/\eta-1}[1+\delta(1-x^{\rho/\eta})]^{-1/\eta-1}[1+\delta(1-(1+\rho/\eta)x^{\rho/\eta})] $$ for all \(x>1\) when parametrized by \(\rho\).

The control argument is a list that can supply any of the following components:

upperbound

The upperbound used in the optimize function when computing numerical quantiles, default to 1e6.

tol

the desired accuracy used in the optimize function when computing numerical quantiles, default to 1e-9.

References

J. Beirlant, E. Joossens, J. Segers (2009), Second-order refined peaks-over-threshold modelling for heavy-tailed distributions, Journal of Statistical Planning and Inference, Volume 139, Issue 8, Pages 2800-2815.

C. Dutang, Y. Goegebeur, A. Guillou (2014), Robust and bias-corrected estimation of the coefficient of tail dependence, Insurance: Mathematics and Economics

This work was supported by a research grant (VKR023480) from VILLUM FONDEN and an international project for scientific cooperation (PICS-6416).

Examples

#####
# (1) density function
x <- seq(0, 5, length=24)

cbind(x, dEPD(x, 1/2, 1/4, -1))

#####
# (2) distribution function

cbind(x, pEPD(x, 1/2, 1/4, -1, lower=FALSE))