triangular: Triangular distribution

In VaRES: Computes Value at Risk and Expected Shortfall for over 100 Parametric Distributions

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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/03610918.2014.944658")}

Examples

x=runif(10,min=0,max=1)
dtriangular(x)
ptriangular(x)
vartriangular(x)
estriangular(x)

VaRES documentation built on April 22, 2023, 1:16 a.m.