pareto function

Pareto distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Pareto distribution due to Pareto (1964) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = c K^c x^{-c - 1},\\&\displaystyleF (x) = 1 - \left( \frac {K}{x} \right)^c,\\&\displaystyle{\rm VaR}_p (X) = K (1 - p)^{-1 / c},\\&\displaystyle{\rm ES}_p (X) = \frac {K c}{p (1 - c)} (1 - p)^{1 - 1 / c} - \frac {K c}{p (1 - c)}\end{array} [REMOVE_ME_2]$

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

Description

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

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

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

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)
dpareto(x)
ppareto(x)
varpareto(x)
espareto(x)