rgamma function

Reflected gamma distribution

Computes the pdf, cdf, value at risk and expected shortfall for the reflected gamma distribution due to Borgi (1965) given by [REMOVE_ME] \begin{array}{ll}&\displaystylef (x) = \frac {1}{\displaystyle 2 \phi \Gamma \left( a \right)}\left | \frac {x - \theta}{\phi} \right |^{a - 1}\exp \left\{ -\left | \frac {x - \theta}{\phi} \right | \right\},\\&\displaystyleF (x) =\left\{\begin{array}{ll}\displaystyle\frac {1}{2} Q \left( a, \frac {\theta - x}{\phi} \right), &\mbox{if $x \leq \theta$,}\\\\\displaystyle1 - \frac {1}{2} Q \left( a, \frac {x - \theta}{\phi} \right), &\mbox{if $x > \theta$,}\end{array}\right.\\&\displaystyle{\rm VaR}_p (X) =\left\{\begin{array}{ll}\displaystyle\theta - \phi Q^{-1} \left( a, 2 p \right), &\mbox{if $p \leq 1/2$,}\\\\\displaystyle\theta + \phi Q^{-1} \left( a, 2 (1 - p) \right), &\mbox{if $p > 1/2$,}\end{array}\right.\\&\displaystyle{\rm ES}_p (X) =\left\{\begin{array}{ll}\displaystyle\theta - \frac {\phi}{p} \int_0^p Q^{-1} \left( a, 2 v \right) dv, &\mbox{if $p \leq 1/2$,}\\\\\displaystyle\theta - \frac {\phi}{p} \int_0^{1/2} Q^{-1} \left( a, 2 v \right) dv +\frac {\phi}{p} \int_{1/2}^p Q^{-1}\left( a, 2 (1 - v) \right) dv, & \mbox{if $p > 1/2$}\end{array}\right.\end{array} [REMOVE_ME_2]

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \theta < \infty$, the location parameter, $\phi > 0$, the scale parameter, and $a > 0$, the shape parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the reflected gamma distribution due to Borgi (1965) given by

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \theta < \infty$, the location parameter, $\phi > 0$, the scale parameter, and $a > 0$, the shape parameter.

drgamma(x, a=1, theta=0, phi=1, log=FALSE)
prgamma(x, a=1, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)
varrgamma(p, a=1, theta=0, phi=1, log.p=FALSE, lower.tail=TRUE)
esrgamma(p, a=1, theta=0, phi=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
• theta: the value of the location parameter, can take any real value, the default is zero
• phi: the value of the scale parameter, must be positive, the default is 1
• 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

Returns

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.

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, tools:::Rd_expr_doi("10.1080/03610918.2014.944658")

Author(s)

Saralees Nadarajah

Examples

x=runif(10,min=0,max=1)
drgamma(x)
prgamma(x)
varrgamma(x)
esrgamma(x)