# kum: Kumaraswamy distribution In VaRES: Computes value at risk and expected shortfall for over 100 parametric distributions

## 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} ≤ft( 1 - x^a \right)^{b - 1}, \\ &\displaystyle F (x) = 1 - ≤ft( 1 - x^a \right)^b, \\ &\displaystyle {\rm VaR}_p (X) = ≤ft[ 1 - (1 - p)^{1 / b} \right]^{1 / a}, \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{p} \int_0^p ≤ft[ 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

 1 2 3 4 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)

 1 2 3 4 5 x=runif(10,min=0,max=1) dkum(x) pkum(x) varkum(x) eskum(x)