EPdist: The Extended Pareto Distribution In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

Description

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

Usage

 1 2 3 4 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 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 FALSE. lower.tail Logical indicating if the probabilities are of the form P(X≤ 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+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.

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.

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.  