Generalized Pareto distribution

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Description

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

\begin{array}{ll} &\displaystyle f (x) = \frac {1}{k} ≤ft( 1 - \frac {c x}{k} \right)^{1 / c - 1}, \\ &\displaystyle F (x) = 1 - ≤ft( 1 - \frac {c x}{k} \right)^{1 / c}, \\ &\displaystyle {\rm VaR}_p (X) = \frac {k}{c} ≤ft[ 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}

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 -∞ < c < ∞, the shape parameter.

Usage

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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

Value

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.

Author(s)

Saralees Nadarajah

References

S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall, submitted

Examples

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