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.

`Pareto`, `GPD`, `Distributions`

Examples

 ```1 2 3 4 5 6 7``` ```# 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") ```

Example output

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ReIns documentation built on July 2, 2020, 4:03 a.m.