| triangular {VaRES} | R Documentation |
Triangular distribution
Description
Computes the pdf, cdf, value at risk and expected shortfall for the triangular distribution given by
\begin{array}{ll}
&\displaystyle
f (x) = \left\{
\begin{array}{ll}
\displaystyle
0, & \mbox{if $x < a$,}
\\
\\
\displaystyle
\frac {2 (x - a)}{(b - a) (c - a)}, & \mbox{if $a \leq x \leq c$,}
\\
\\
\displaystyle
\frac {2 (b - x)}{(b - a) (b - c)}, & \mbox{if $c < x \leq b$,}
\\
\\
\displaystyle
0, & \mbox{if $b < x$,}
\end{array} \right.
\\
&\displaystyle
F (x) = \left\{
\begin{array}{ll}
\displaystyle
0, & \mbox{if $x < a$,}
\\
\\
\displaystyle
\frac {(x - a)^2}{(b - a) (c - a)}, & \mbox{if $a \leq x \leq c$,}
\\
\\
\displaystyle
1 - \frac {(b - x)^2}{(b - a) (b - c)}, & \mbox{if $c < x \leq b$,}
\\
\\
\displaystyle
1, & \mbox{if $b < x$,}
\end{array} \right.
\\
&\displaystyle
{\rm VaR}_p (X) = \left\{
\begin{array}{ll}
\displaystyle
a + \sqrt{p (b - a) (c - a)}, & \mbox{if $0 < p < \frac {c - a}{b - a}$,}
\\
\\
\displaystyle
b - \sqrt{(1 - p) (b - a) (b - c)}, & \mbox{if $\frac {c - a}{b - a} \leq p < 1$,}
\end{array} \right.
\\
&\displaystyle
{\rm ES}_p (X) =
\left\{
\begin{array}{ll}
\displaystyle
a + \frac {2}{3} \sqrt{p (b - a) (c - a)}, & \mbox{if $0 < p < \frac {c - a}{b - a}$,}
\\
\\
\displaystyle
b + \frac {a - c}{p} + \frac {2 (2 c - a - b)}{3 p} +2 \sqrt{(b - a) (b - c)} \frac {(1 - p)^{3/2}}{3 p}, &
\mbox{if $\frac {c - a}{b - a} \leq p < 1$}
\end{array} \right.
\end{array}
for a \leq x \leq b, 0 < p < 1, -\infty < a < \infty, the first location parameter, -\infty < a < c < \infty, the second location parameter, and -\infty < c < b < \infty, the third location parameter.
Usage
dtriangular(x, a=0, b=2, c=1, log=FALSE)
ptriangular(x, a=0, b=2, c=1, log.p=FALSE, lower.tail=TRUE)
vartriangular(p, a=0, b=2, c=1, log.p=FALSE, lower.tail=TRUE)
estriangular(p, a=0, b=2, c=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 |
a |
the value of the first location parameter, can take any real value, the default is zero |
c |
the value of the second location parameter, can take any real value but must be greater than a, the default is 1 |
b |
the value of the third location parameter, can take any real value but must be greater than c, the default is 2 |
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)
dtriangular(x)
ptriangular(x)
vartriangular(x)
estriangular(x)