| tsp {VaRES} | R Documentation |
Two sided power distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the two sided power distribution due to van Dorp and Kotz (2002) given by
\begin{array}{ll}
&\displaystyle
f (x) = \left\{ \begin{array}{ll}
\displaystyle
a \left( \frac {x}{\theta} \right)^{a - 1}, & \mbox{if $0 < x \leq \theta$,}
\\
\displaystyle
a \left( \frac {1 - x}{1 - \theta} \right)^{a - 1}, & \mbox{if $\theta < x < 1$,}
\end{array}
\right.
\\
&\displaystyle
F (x) = \left\{ \begin{array}{ll}
\displaystyle
\theta \left( \frac {x}{\theta} \right)^a, & \mbox{if $0 < x \leq \theta$,}
\\
\displaystyle
1 - (1 - \theta) \left( \frac {1 - x}{1 - \theta} \right)^a, & \mbox{if $\theta < x < 1$,}
\end{array}
\right.
\\
&\displaystyle
{\rm VaR}_p (X) = \left\{ \begin{array}{ll}
\displaystyle
\theta \left( \frac {p}{\theta} \right)^{1 / a}, & \mbox{if $0 < p \leq \theta$,}
\\
\displaystyle
1 - (1 - \theta) \left( \frac {1 - p}{1 - \theta} \right)^{1 / a}, & \mbox{if $\theta < p < 1$,}
\end{array}
\right.
\\
&\displaystyle
{\rm ES}_p (X) = \left\{ \begin{array}{ll}
\displaystyle
\frac {a \theta}{a + 1} \left( \frac {p}{\theta} \right)^{1 / a}, & \mbox{if $0 < p \leq \theta$,}
\\
\displaystyle
1 - \frac {\theta}{p} + \frac {a (2 \theta - 1)}{(a + 1) p} + \frac {a (1 - \theta)^2}{(a + 1) p}
\left( \frac {1 - p}{1 - \theta} \right)^{1 + 1 / a}, & \mbox{if $\theta < p < 1$}
\end{array}
\right.
\end{array}
for 0 < x < 1, 0 < p < 1, a > 0, the shape parameter, and -\infty < \theta < \infty, the location parameter.
Usage
dtsp(x, a=1, theta=0.5, log=FALSE)
ptsp(x, a=1, theta=0.5, log.p=FALSE, lower.tail=TRUE)
vartsp(p, a=1, theta=0.5, log.p=FALSE, lower.tail=TRUE)
estsp(p, a=1, theta=0.5)
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, must take a value in the unit interval, the default is 0.5 |
a |
the value of the shape 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 |
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)
dtsp(x)
ptsp(x)
vartsp(x)
estsp(x)