tGPD: The truncated generalised Pareto distribution In ReIns: Functions from "Reinsurance: Actuarial and Statistical Aspects"

Description

Density, distribution function, quantile function and random generation for the truncated Generalised Pareto Distribution (GPD).

Usage

 ```1 2 3 4``` ```dtgpd(x, gamma, mu = 0, sigma, endpoint = Inf, log = FALSE) ptgpd(x, gamma, mu = 0, sigma, endpoint = Inf, lower.tail = TRUE, log.p = FALSE) qtgpd(p, gamma, mu = 0, sigma, endpoint = Inf, lower.tail = TRUE, log.p = FALSE) rtgpd(n, gamma, mu = 0, sigma, endpoint = Inf) ```

Arguments

 `x` Vector of quantiles. `p` Vector of probabilities. `n` Number of observations. `gamma` The γ parameter of the GPD, a real number. `mu` The μ parameter of the GPD, a strictly positive number. Default is 0. `sigma` The σ parameter of the GPD, a strictly positive number. `endpoint` Endpoint of the truncated GPD. The default value is `Inf` for which the truncated GPD corresponds to the ordinary GPD. `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 truncated GPD is equal to F_T(x) = F(x) / F(T) for x ≤ T where F is the CDF of the ordinary GPD and T is the endpoint (truncation point) of the truncated GPD.

Value

`dtgpd` gives the density function evaluated in x, `ptgpd` the CDF evaluated in x and `qtgpd` the quantile function evaluated in p. The length of the result is equal to the length of x or p.

`rtgpd` returns a random sample of length n.

Author(s)

Tom Reynkens

`tGPD`, `Pareto`, `Distributions`
 ```1 2 3 4 5 6 7``` ```# Plot of the PDF x <- seq(0, 10, 0.01) plot(x, dtgpd(x, gamma=1/2, sigma=5, endpoint=8), xlab="x", ylab="PDF", type="l") # Plot of the CDF x <- seq(0, 10, 0.01) plot(x, ptgpd(x, gamma=1/2, sigma=5, endpoint=8), xlab="x", ylab="CDF", type="l") ```