power1 {VaRES}

Power function I distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the power function I distribution given by

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

for 0 < x < 1, 0 < p < 1, and a > 0, the shape parameter.

Usage

dpower1(x, a=1, log=FALSE)
ppower1(x, a=1, log.p=FALSE, lower.tail=TRUE)
varpower1(p, a=1, log.p=FALSE, lower.tail=TRUE)
espower1(p, a=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

a

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)
dpower1(x)
ppower1(x)
varpower1(x)
espower1(x)