pareto {VaRES}

Pareto distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Pareto distribution due to Pareto (1964) given by

\begin{array}{ll} &\displaystyle f (x) = c K^c x^{-c - 1}, \\ &\displaystyle F (x) = 1 - \left( \frac {K}{x} \right)^c, \\ &\displaystyle {\rm VaR}_p (X) = K (1 - p)^{-1 / c}, \\ &\displaystyle {\rm ES}_p (X) = \frac {K c}{p (1 - c)} (1 - p)^{1 - 1 / c} - \frac {K c}{p (1 - c)} \end{array}

for x \geq K, 0 < p < 1, K > 0, the scale parameter, and c > 0, the shape parameter.

Usage

dpareto(x, K=1, c=1, log=FALSE)
ppareto(x, K=1, c=1, log.p=FALSE, lower.tail=TRUE)
varpareto(p, K=1, c=1, log.p=FALSE, lower.tail=TRUE)
espareto(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, must be positive, 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

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, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dpareto(x)
ppareto(x)
varpareto(x)
espareto(x)