Gamma {VaRES}

Gamma distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the gamma distribution given by

\begin{array}{ll} &\displaystyle f (x) = \frac {b^a x^{a - 1} \exp (-b x)}{\Gamma (a)}, \\ &\displaystyle F (x) = \frac {\gamma (a, b x)}{\Gamma (a)}, \\ &\displaystyle {\rm VaR}_p (X) = \frac {1}{b} Q^{-1} (a, 1 - p), \\ &\displaystyle {\rm ES}_p (X) = \frac {1}{b p} \int_0^p Q^{-1} (a, 1 - v) dv \end{array}

for x > 0, 0 < p < 1, b > 0, the scale parameter, and a > 0, the shape parameter, where \gamma (a, x) = \int_0^x t^{a - 1} \exp \left( -t \right) dt denotes the incomplete gamma function, Q (a, x) = \int_x^\infty t^{a - 1} \exp \left( -t \right) dt / \Gamma (a) denotes the regularized complementary incomplete gamma function, \Gamma (a) = \int_0^\infty t^{a - 1} \exp \left( -t \right) dt denotes the gamma function, and Q^{-1} (a, x) denotes the inverse of Q (a, x).

Usage

dGamma(x, a=1, b=1, log=FALSE)
pGamma(x, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varGamma(p, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
esGamma(p, a=1, b=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

b

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

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)
dGamma(x)
pGamma(x)
varGamma(x)
esGamma(x)