| PCTAlaplace {VaRES} | R Documentation |
Poiraud-Casanova-Thomas-Agnan Laplace distribution
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
\begin{array}{ll}
&\displaystyle
f (x) = \left\{
\begin{array}{ll}
\displaystyle
a \left( 1 - a \right) \exp \left\{ \left( 1 - a \right)
\left( x - \theta \right) \right\}, & \mbox{if $x \leq \theta$,}
\\
\\
\displaystyle
a \left( 1 - a \right) \exp \left\{ a \left( \theta - x \right) \right\}, & \mbox{if $x > \theta$,}
\end{array}
\right.
\\
&\displaystyle
F (x) =
\left\{
\begin{array}{ll}
\displaystyle
a \exp \left\{ \left( 1 - a \right)
\left( x - \theta \right) \right\}, & \mbox{if $x \leq \theta$,}
\\
\\
\displaystyle
1 - \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}
for -\infty < x < \infty, 0 < p < 1, -\infty < \theta < \infty, the location parameter, and
a > 0, the scale parameter.
Usage
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 |
Value
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.
Author(s)
Saralees Nadarajah
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, doi:10.1080/03610918.2014.944658
Examples
x=runif(10,min=0,max=1)
dPCTAlaplace(x)
pPCTAlaplace(x)
varPCTAlaplace(x)
esPCTAlaplace(x)