laplace function

Laplace distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Laplace distribution due to due to Laplace (1774) given by [REMOVE_ME] \begin{array}{ll}&\displaystylef (x) = \frac {1}{2 \sigma} \exp \left( -\frac {\mid x - \mu \mid}{\sigma} \right),\\&\displaystyleF (x) = \left\{\begin{array}{ll}\displaystyle\frac {1}{2} \exp \left( \frac {x - \mu}{\sigma} \right), & \mbox{if $x < \mu$,}\\\\\displaystyle1 - \frac {1}{2} \exp \left( -\frac {x - \mu}{\sigma} \right), & \mbox{if $x \geq \mu$,}\end{array}\right.\\&\displaystyle{\rm VaR}_p (X) = \left\{\begin{array}{ll}\displaystyle\mu + \sigma \log (2 p), & \mbox{if $p < 1/2$,}\\\\\displaystyle\mu - \sigma \log \left[ 2 (1 - p) \right], & \mbox{if $p \geq 1/2$,}\end{array}\right.\\&\displaystyle{\rm ES}_p (X) =\left\{\begin{array}{ll}\displaystyle\mu + \sigma \left[ \log (2 p) - 1 \right], & \mbox{if $p < 1/2$,}\\\\\displaystyle\mu + \sigma - \frac {\sigma}{p} + \sigma \frac {1 - p}{p} \log (1 - p) +\sigma \frac {1 - p}{p} \log 2, & \mbox{if $p \geq 1/2$}\end{array}\right.\end{array} [REMOVE_ME_2]

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \mu < \infty$, the location parameter, and $\sigma > 0$, the scale parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Laplace distribution due to due to Laplace (1774) given by

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \mu < \infty$, the location parameter, and $\sigma > 0$, the scale parameter.

dlaplace(x, mu=0, sigma=1, log=FALSE)
plaplace(x, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
varlaplace(p, mu=0, sigma=1, log.p=FALSE, lower.tail=TRUE)
eslaplace(p, mu=0, sigma=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
• mu: the value of the location parameter, can take any real value, the default is zero
• sigma: the value of the 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)
dlaplace(x)
plaplace(x)
varlaplace(x)
eslaplace(x)