PCTAlaplace function

Poiraud-Casanova-Thomas-Agnan Laplace distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Poiraud-Casanova-Thomas-Agnan Laplace distribution due to Poiraud-Casanova and Thomas-Agnan (2000) given by [REMOVE_ME] \begin{array}{ll}&\displaystylef (x) = \left\{\begin{array}{ll}\displaystylea \left( 1 - a \right) \exp \left\{ \left( 1 - a \right)\left( x - \theta \right) \right\}, & \mbox{if $x \leq \theta$,}\\\\\displaystylea \left( 1 - a \right) \exp \left\{ a \left( \theta - x \right) \right\}, & \mbox{if $x > \theta$,}\end{array}\right.\\&\displaystyleF (x) =\left\{\begin{array}{ll}\displaystylea \exp \left\{ \left( 1 - a \right)\left( x - \theta \right) \right\}, & \mbox{if $x \leq \theta$,}\\\\\displaystyle1 - \left( 1 - a \right)\exp \left\{ a \left( \theta - x \right) \right\}, & \mbox{if $x > \theta$,}\end{array}\right.\\&\displaystyle{\rm VaR}_p (X) = \left\{\begin{array}{ll}\displaystyle\theta + \frac {1}{1 - a} \log \left( \frac {p}{a} \right), & \mbox{if $p \leq a$,}\\\\\displaystyle\theta - \frac {1}{a} \log \left( \frac {1 - p}{1 - a} \right), & \mbox{if $p > a$,}\end{array}\right.\\&\displaystyle{\rm ES}_p (X) =\left\{\begin{array}{ll}\displaystyle\theta - \frac {\log a}{1 - a} + \frac {\log p - 1}{(1 - a) p}, & \mbox{if $p \leq a$,}\\\\\displaystyle\theta - \frac {1}{a} + \frac {1}{p} - \frac {a}{(1 - a) p} + \frac {1 - p}{a p}\log \left( \frac {1 - p}{1 - a} \right), & \mbox{if $p > a$}\end{array}\right.\end{array} [REMOVE_ME_2]

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

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Poiraud-Casanova-Thomas-Agnan Laplace distribution due to Poiraud-Casanova and Thomas-Agnan (2000) given by

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

dPCTAlaplace(x, a=0.5, theta=0, log=FALSE)
pPCTAlaplace(x, a=0.5, theta=0, log.p=FALSE, lower.tail=TRUE)
varPCTAlaplace(p, a=0.5, theta=0, log.p=FALSE, lower.tail=TRUE)
esPCTAlaplace(p, a=0.5, 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
• a: the value of the scale parameter, must be in the unit interval, the default is 0.5
• 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)
dPCTAlaplace(x)
pPCTAlaplace(x)
varPCTAlaplace(x)
esPCTAlaplace(x)