moexp function

Marshall-Olkin exponential distribution

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

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

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Marshall-Olkin exponential distribution due to Marshall and Olkin (1997) given by

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

dmoexp(x, lambda=1, a=1, log=FALSE)
pmoexp(x, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
varmoexp(p, lambda=1, a=1, log.p=FALSE, lower.tail=TRUE)
esmoexp(p, lambda=1, a=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
• 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)
dmoexp(x)
pmoexp(x)
varmoexp(x)
esmoexp(x)