moweibull function

Marshall-Olkin Weibull distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Marshall-Olkin Weibull distribution due to Marshall and Olkin (1997) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef(x) = b \lambda^b x^{b - 1} \exp \left[ (\lambda x)^b \right]\left\{ \exp \left[ (\lambda x)^b \right] - 1 + a \right\}^{-2},\\&\displaystyleF(x) = \frac {\displaystyle \exp \left[ (\lambda x)^b \right] - 2 + a}{\displaystyle \exp \left[ (\lambda x)^b \right] - 1 + a},\\&\displaystyle{\rm VaR}_p (X) = \frac {1}{\lambda} \left[ \log \left( \frac {1}{1 - p} + 1 - a \right) \right]^{1 / b},\\&\displaystyle{\rm ES}_p (X) = \frac {1}{\lambda p} \int_0^p \left[ \log \left( \frac {1}{1 - v} + 1 - a \right) \right]^{1 / b} dv\end{array} [REMOVE_ME_2]$

for $x > 0$, $0 < p < 1$, $a > 0$, the first scale parameter, $b > 0$, the shape parameter, and $\lambda > 0$, the second scale parameter.

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

for $x > 0$, $0 < p < 1$, $a > 0$, the first scale parameter, $b > 0$, the shape parameter, and $\lambda > 0$, the second scale parameter.

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

Returns

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.

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, tools:::Rd_expr_doi("10.1080/03610918.2014.944658")

Author(s)

Saralees Nadarajah

Examples

x=runif(10,min=0,max=1)
dmoweibull(x)
pmoweibull(x)
varmoweibull(x)
esmoweibull(x)