# EPD: The Extended Pareto Distribution In RTDE: Robust Tail Dependence Estimation

## Description

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

## Usage

 ```1 2 3 4 5``` ```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) ```

## 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 ≤ x], otherwise, P[X > x]. `control` A list of control paremeters. See section Details.

## Details

The extended Pareto distribution is defined by the following density

f(x) = 1/η x^{-1/η-1}[1+δ(1-x^{-τ})]^{-1/η-1}[1+δ(1-(1-τ)x^{-τ})]

for all x>1 when parametrized by τ. However, a typical parametrization is obtained by setting τ=-ρ/η, i.e.

f(x) = 1/η x^{-1/η-1}[1+δ(1-x^{ρ/η})]^{-1/η-1}[1+δ(1-(1+ρ/η)x^{ρ/η})]

for all x>1 when parametrized by ρ.

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`.

## 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.

## Author(s)

Christophe Dutang

## 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 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```##### # (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)) ```

RTDE documentation built on Jan. 8, 2020, 5:09 p.m.