genpareto function

Generalized Pareto distribution

Computes the pdf, cdf, value at risk and expected shortfall for the generalized Pareto distribution due to Pickands (1975) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {1}{k} \left( 1 - \frac {c x}{k} \right)^{1 / c - 1},\\&\displaystyleF (x) = 1 - \left( 1 - \frac {c x}{k} \right)^{1 / c},\\&\displaystyle{\rm VaR}_p (X) = \frac {k}{c} \left[ 1 - (1 - p)^c \right],\\&\displaystyle{\rm ES}_p (X) = \frac {k}{c} + \frac {k (1 - p)^{c + 1}}{p c (c + 1)} - \frac {k}{p c (c + 1)}\end{array} [REMOVE_ME_2]$

for $x < k/c$ if $c > 0$, $x > k/c$ if $c < 0$, $x > 0$ if $c = 0$, $0 < p < 1$, $k > 0$, the scale parameter and $-\infty < c < \infty$, the shape parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the generalized Pareto distribution due to Pickands (1975) given by

for $x < k/c$ if $c > 0$, $x > k/c$ if $c < 0$, $x > 0$ if $c = 0$, $0 < p < 1$, $k > 0$, the scale parameter and $-\infty < c < \infty$, the shape parameter.

dgenpareto(x, k=1, c=1, log=FALSE)
pgenpareto(x, k=1, c=1, log.p=FALSE, lower.tail=TRUE)
vargenpareto(p, k=1, c=1, log.p=FALSE, lower.tail=TRUE)
esgenpareto(p, k=1, c=1)

Arguments

• x: scaler or vector of values at which the pdf or cdf needs to be computed
• p: scaler or vector of values at which the value at risk or expected shortfall needs to be computed
• k: the value of the scale parameter, must be positive, the default is 1
• c: the value of the shape parameter, can take any real value, the default is 1
• log: if TRUE then log(pdf) are returned
• log.p: if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
• lower.tail: if FALSE then 1-cdf are returned and quantiles are computed for 1-p

Returns

An object of the same length as x, giving the pdf or cdf values computed at x or an object of the same length as p, giving the values at risk or expected shortfall computed at p.

References

Stephen Chan, Saralees Nadarajah & Emmanuel Afuecheta (2016). An R Package for Value at Risk and Expected Shortfall, Communications in Statistics - Simulation and Computation, 45:9, 3416-3434, tools:::Rd_expr_doi("10.1080/03610918.2014.944658")

Author(s)

Saralees Nadarajah

Examples

x=runif(10,min=0,max=1)
dgenpareto(x)
pgenpareto(x)
vargenpareto(x)
esgenpareto(x)