# 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 $$ \begin{array}{ll}&\displaystylef (x) = \left\{\begin{array}{ll}\displaystyle\frac {1}{2 \psi} \exp \left( \frac {x - \theta}{\psi} \right), & \mbox{if $x \leq \theta$,}\\\\\displaystyle\frac {1}{2 \phi} \exp \left( \frac {\theta - x}{\phi} \right), & \mbox{if $x > \theta$,}\end{array}\right.\\&\displaystyleF (x) = \left\{\begin{array}{ll}\displaystyle\frac {1}{2} \exp \left( \frac {x - \theta}{\psi} \right), & \mbox{if $x \leq \theta$,}\\\\\displaystyle1 - \frac {1}{2} \exp \left( \frac {\theta - x}{\phi} \right), & \mbox{if $x > \theta$,}\end{array}\right.\\&\displaystyle{\rm VaR}_p (X) = \left\{\begin{array}{ll}\displaystyle\theta + \psi \log (2 p), & \mbox{if $p \leq 1/2$,}\\\\\displaystyle\theta - \phi \log \left( 2 (1 - p) \right), & \mbox{if $p > 1/2$,}\end{array}\right.\\&\displaystyle{\rm ES}_p (X) = \left\{\begin{array}{ll}\displaystyle\psi + \theta \log (2 p) - \theta p, & \mbox{if $p \leq 1/2$,}\\\\\displaystyle\theta + \phi + \frac {\psi - \phi - 2 \theta}{2 p} + \frac {\phi}{p} \log 2 - \phi \log 2\\\displaystyle\quad+\frac {\phi}{p} \log (1 - p) - \phi \log (1 - p), & \mbox{if $p > 1/2$}\end{array}\right.\end{array} $$ 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. ```r 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 ```r x=runif(10,min=0,max=1) dMlaplace(x) pMlaplace(x) varMlaplace(x) esMlaplace(x) ```

Mlaplace function