VaRES (version 1.0.2)

moweibull: 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)

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.

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

Author

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

Examples

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