VaRES (version 1.0.2)

Mlaplace: McGill Laplace distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the McGill Laplace distribution due to McGill (1962) given by $$\begin{array}{ll} &\displaystyle f (x) = \left\{ \begin{array}{ll} \displaystyle \frac {1}{2 \psi} \exp \left( \frac {x - \theta}{\psi} \right), & \mbox{if $x \leq \theta$,} \\ \\ \displaystyle \frac {1}{2 \phi} \exp \left( \frac {\theta - x}{\phi} \right), & \mbox{if $x > \theta$,} \end{array} \right. \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle \frac {1}{2} \exp \left( \frac {x - \theta}{\psi} \right), & \mbox{if $x \leq \theta$,} \\ \\ \displaystyle 1 - \frac {1}{2} \exp \left( \frac {\theta - x}{\phi} \right), & \mbox{if $x > \theta$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle \theta + \psi \log (2 p), & \mbox{if $p \leq 1/2$,} \\ \\ \displaystyle \theta - \phi \log \left( 2 (1 - p) \right), & \mbox{if $p > 1/2$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle \psi + \theta \log (2 p) - \theta p, & \mbox{if $p \leq 1/2$,} \\ \\ \displaystyle \theta + \phi + \frac {\psi - \phi - 2 \theta}{2 p} + \frac {\phi}{p} \log 2 - \phi \log 2 \\ \displaystyle \quad +\frac {\phi}{p} \log (1 - p) - \phi \log (1 - p), & \mbox{if $p > 1/2$} \end{array} \right. \end{array}$$ for \(-\infty < x < \infty\), \(0 < p < 1\), \(-\infty < \theta < \infty\), the location parameter, \(\phi > 0\), the first scale parameter, and \(\psi > 0\), the second scale parameter.

Usage

dMlaplace(x, theta=0, phi=1, psi=1, log=FALSE)
pMlaplace(x, theta=0, phi=1, psi=1, log.p=FALSE, lower.tail=TRUE)
varMlaplace(p, theta=0, phi=1, psi=1, log.p=FALSE, lower.tail=TRUE)
esMlaplace(p, theta=0, phi=1, psi=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

theta

the value of the location parameter, can take any real value, the default is zero

phi

the value of the first scale parameter, must be positive, the default is 1

psi

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

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)
dMlaplace(x)
pMlaplace(x)
varMlaplace(x)
esMlaplace(x)