asylaplace function

Asymmetric Laplace distribution

Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric Laplace distribution due to Kotz et al. (2001) given by [REMOVE_ME] \begin{array}{ll}&\displaystyle\displaystylef(x) = \left\{ \begin{array}{ll}\displaystyle\frac {\kappa \sqrt{2}}{\tau \left( 1 + \kappa^2 \right)}\exp \left( -\frac {\kappa \sqrt{2}}{\tau} \left | x - \theta \right | \right), & \mbox{if $x \geq \theta$,}\\\\\displaystyle\frac {\kappa \sqrt{2}}{\tau \left( 1 + \kappa^2 \right)}\exp \left( -\frac {\sqrt{2}}{\kappa \tau} \left | x - \theta \right | \right), & \mbox{if $x < \theta$,}\end{array}\right.\\&\displaystyleF (x) = \left\{ \begin{array}{ll}\displaystyle1 - \frac {1}{1 + \kappa^2} \exp \left( \frac {\kappa \sqrt{2} (\theta - x)}{\tau} \right), &\mbox{if $x \geq \theta$,}\\\\\displaystyle\frac {\kappa^2}{1 + \kappa^2} \exp \left( \frac {\sqrt{2} (x - \theta)}{\kappa \tau} \right), & \mbox{if $x < \theta$,}\end{array}\right.\\&\displaystyle{\rm VaR}_p (X) = \left\{ \begin{array}{ll}\displaystyle\theta - \frac {\tau}{\sqrt{2} \kappa}\log \left[ (1 - p) \left( 1 + \kappa^2 \right) \right], & \mbox{if $p \geq \frac {\kappa^2}{1 + \kappa^2}$,}\\\\\displaystyle\theta + \frac {\kappa \tau}{\sqrt{2}} \log \left[ p \left( 1 + \kappa^{-2} \right) \right], &\mbox{if $p < \frac {\kappa^2}{1 + \kappa^2}$,}\end{array}\right.\\&\displaystyle{\rm ES}_p (X) = \left\{ \begin{array}{ll}\displaystyle\frac {\theta}{p} + \theta - \frac {\tau}{\sqrt{2} \kappa} \log \left( 1 + \kappa^2 \right) +\frac {\sqrt{2} \tau \left( 1 + 2 \kappa^2 \right)}{2 \kappa \left( 1 + \kappa^2 \right) p}\log \left( 1 + \kappa^2 \right)\\\displaystyle\quad-\frac {\sqrt{2} \tau \kappa \log \kappa}{\left( 1 + \kappa^2 \right) p} -\frac {\theta \kappa^2}{\left( 1 + \kappa^2 \right) p} +\frac {\tau \left( 1 - \kappa^4 \right)}{\sqrt{2} \kappa \left( 1 + \kappa^2 \right) p}\\\displaystyle\quad-\frac {\tau (1 - p)}{\sqrt{2} \kappa p} + \frac {\tau (1 - p)}{\sqrt{2} \kappa p} \log (1 - p), &\mbox{if $p \geq \frac {\kappa^2}{1 + \kappa^2}$,}\\\\\displaystyle\theta + \frac {\kappa \tau}{\sqrt{2}}\log \left( 1 + \kappa^{-2} \right) +\frac {\kappa \tau}{\sqrt{2}} (\log p - 1), & \mbox{if $p < \frac {\kappa^2}{1 + \kappa^2}$}\end{array}\right.\end{array} [REMOVE_ME_2]

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \theta < \infty$, the location parameter, $\kappa > 0$, the first scale parameter, and $\tau > 0$, the second scale parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric Laplace distribution due to Kotz et al. (2001) given by

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \theta < \infty$, the location parameter, $\kappa > 0$, the first scale parameter, and $\tau > 0$, the second scale parameter.

dasylaplace(x, tau=1, kappa=1, theta=0, log=FALSE)
pasylaplace(x, tau=1, kappa=1, theta=0, log.p=FALSE, lower.tail=TRUE)
varasylaplace(p, tau=1, kappa=1, theta=0, log.p=FALSE, lower.tail=TRUE)
esasylaplace(p, tau=1, kappa=1, theta=0)

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
• kappa: the value of the first scale parameter, must be positive, the default is 1
• tau: 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)
dasylaplace(x)
pasylaplace(x)
varasylaplace(x)
esasylaplace(x)