| kumhalfnorm {VaRES} | R Documentation |
Kumaraswamy half normal distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy half normal distribution due to Cordeiro et al. (2012c) given by
\begin{array}{ll}
&\displaystyle
f (x) = \frac {2a b}{\sigma} \phi \left( \frac {x}{\sigma} \right) \left[ 2 \Phi \left( \frac {x}{\sigma} \right) - 1 \right]^{a - 1}
\left\{ 1 - \left[ 2 \Phi \left( \frac {x}{\sigma} \right) - 1 \right]^a \right\}^{b - 1},
\\
&\displaystyle
F (x) = 1 - \left\{ 1 - \left[ 2 \Phi \left( \frac {x}{\sigma} \right) - 1 \right]^a \right\}^b,
\\
&\displaystyle
{\rm VaR}_p (X) = \sigma \Phi^{-1} \left( \frac {1}{2} + \frac {1}{2}
\left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} \right),
\\
&\displaystyle
{\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \Phi^{-1} \left( \frac {1}{2} + \frac {1}{2}
\left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} \right) dv
\end{array}
for x > 0, 0 < p < 1,
\sigma > 0, the scale parameter, a > 0, the first shape parameter, and b > 0, the second shape parameter.
Usage
dkumhalfnorm(x, sigma=1, a=1, b=1, log=FALSE)
pkumhalfnorm(x, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varkumhalfnorm(p, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
eskumhalfnorm(p, sigma=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 |
sigma |
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)
dkumhalfnorm(x)
pkumhalfnorm(x)
varkumhalfnorm(x)
eskumhalfnorm(x)