| moweibull {VaRES} | R Documentation |
Marshall-Olkin Weibull distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Marshall-Olkin Weibull distribution due to Marshall and Olkin (1997) given by
\begin{array}{ll}
&\displaystyle
f(x) = b \lambda^b x^{b - 1} \exp \left[ (\lambda x)^b \right]
\left\{ \exp \left[ (\lambda x)^b \right] - 1 + a \right\}^{-2},
\\
&\displaystyle
F(x) = \frac {\displaystyle \exp \left[ (\lambda x)^b \right] - 2 + a}
{\displaystyle \exp \left[ (\lambda x)^b \right] - 1 + a},
\\
&\displaystyle
{\rm VaR}_p (X) = \frac {1}{\lambda} \left[ \log \left( \frac {1}{1 - p} + 1 - a \right) \right]^{1 / b},
\\
&\displaystyle
{\rm ES}_p (X) = \frac {1}{\lambda p} \int_0^p \left[ \log \left( \frac {1}{1 - v} + 1 - a \right) \right]^{1 / b} dv
\end{array}
for x > 0, 0 < p < 1, a > 0, the first scale parameter, b > 0, the shape parameter,
and \lambda > 0, the second scale parameter.
Usage
dmoweibull(x, a=1, b=1, lambda=1, log=FALSE)
pmoweibull(x, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
varmoweibull(p, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
esmoweibull(p, a=1, b=1, lambda=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 scale parameter, must be positive, the default is 1 |
lambda |
the value of the second scale parameter, must be positive, the default is 1 |
b |
the value of the 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)
dmoweibull(x)
pmoweibull(x)
varmoweibull(x)
esmoweibull(x)