Mlaplace function

McGill Laplace distribution

Computes the pdf, cdf, value at risk and expected shortfall for the McGill Laplace distribution due to McGill (1962) given by [REMOVE_ME] \begin{array}{ll}&\displaystylef (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.\\&\displaystyleF (x) = \left\{\begin{array}{ll}\displaystyle\frac {1}{2} \exp \left( \frac {x - \theta}{\psi} \right), & \mbox{if $x \leq \theta$,}\\\\\displaystyle1 - \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} [REMOVE_ME_2]

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.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the McGill Laplace distribution due to McGill (1962) given by

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.

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)

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

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