kumexp {VaRES} | R Documentation |
Kumaraswamy exponential distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy exponential distribution due to Cordeiro and de Castro (2011) given by
\begin{array}{ll}
&\displaystyle
f (x) = a b \lambda \exp (-\lambda x) \left[ 1 - \exp (-\lambda x) \right]^{a - 1}
\left\{ 1 - \left[ 1 - \exp (-\lambda x) \right]^a \right\}^{b - 1},
\\
&\displaystyle
F (x) = 1 - \left\{ 1 - \left[ 1 - \exp (-\lambda x) \right]^a \right\}^b,
\\
&\displaystyle
{\rm VaR}_p (X) = -\frac {1}{\lambda} \log \left\{ 1 - \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} \right\},
\\
&\displaystyle
{\rm ES}_p (X) = -\frac {1}{p \lambda} \int_0^p \log \left\{ 1 - \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} \right\} dv
\end{array}
for x > 0
, 0 < p < 1
, a > 0
, the first shape parameter, b > 0
, the second shape parameter, and \lambda > 0
, the scale parameter.
Usage
dkumexp(x, lambda=1, a=1, b=1, log=FALSE)
pkumexp(x, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varkumexp(p, lambda=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
eskumexp(p, lambda=1, 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 |
lambda |
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 |
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)
dkumexp(x)
pkumexp(x)
varkumexp(x)
eskumexp(x)