Gamma function

Gamma distribution

Computes the pdf, cdf, value at risk and expected shortfall for the gamma distribution given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {b^a x^{a - 1} \exp (-b x)}{\Gamma (a)},\\&\displaystyleF (x) = \frac {\gamma (a, b x)}{\Gamma (a)},\\&\displaystyle{\rm VaR}_p (X) = \frac {1}{b} Q^{-1} (a, 1 - p),\\&\displaystyle{\rm ES}_p (X) = \frac {1}{b p} \int_0^p Q^{-1} (a, 1 - v) dv\end{array} [REMOVE_ME_2]$

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 $Q (a, x)$.

Description

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

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 $Q (a, x)$.

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

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