kumpareto {VaRES}

Kumaraswamy Pareto distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Pareto distribution due to Pereira et al. (2013) given by

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

for x \geq K, 0 < p < 1, K > 0, the scale parameter, c > 0, the first shape parameter, a > 0, the second shape parameter, and b > 0, the third shape parameter.

Usage

dkumpareto(x, K=1, a=1, b=1, c=1, log=FALSE)
pkumpareto(x, K=1, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)
varkumpareto(p, K=1, a=1, b=1, c=1, log.p=FALSE, lower.tail=TRUE)
eskumpareto(p, K=1, a=1, b=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

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

c

the value of the third 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)
dkumpareto(x)
pkumpareto(x)
varkumpareto(x)
eskumpareto(x)