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.
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)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.
scaler or vector of values at which the pdf or cdf needs to be computed
scaler or vector of values at which the value at risk or expected shortfall needs to be computed
the value of the first location parameter, can take any real value, the default is zero
the value of the second location parameter, can take any real value but must be greater than a, the default is 1
the value of the third location parameter, can take any real value but must be greater than c, the default is 2
if TRUE then log(pdf) are returned
if TRUE then log(cdf) are returned and quantiles are computed for exp(p)
if FALSE then 1-cdf are returned and quantiles are computed for 1-p
Saralees Nadarajah
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")
x=runif(10,min=0,max=1)
dtriangular(x)
ptriangular(x)
vartriangular(x)
estriangular(x)Run the code above in your browser using DataLab