kum {VaRES}

Kumaraswamy distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy distribution due to Kumaraswamy (1980) given by

\begin{array}{ll} &\displaystyle f (x) = a b x^{a - 1} \left( 1 - x^a \right)^{b - 1}, \\ &\displaystyle F (x) = 1 - \left( 1 - x^a \right)^b, \\ &\displaystyle {\rm VaR}_p (X) = \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} dv \end{array}

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

Usage

dkum(x, a=1, b=1, log=FALSE)
pkum(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varkum(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
eskum(p, a=1, b=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 first shape parameter, must be positive, the default is 1

b

the value of the second 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)
dkum(x)
pkum(x)
varkum(x)
eskum(x)