staceygamma function

Stacy distribution

Computes the pdf, cdf, value at risk and expected shortfall for Stacy distribution due to Stacy (1962) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {c x^{c \gamma - 1} \exp \left[ -(x / \theta)^c \right]}{\theta^{c \gamma} \Gamma (\gamma)},\\&\displaystyleF (x) = 1 - Q \left( \gamma, \left( \frac {x}{\theta} \right)^c \right),\\&\displaystyle{\rm VaR}_p (X) = \theta \left[ Q^{-1} (\gamma, 1 - p) \right]^{1 / c},\\&\displaystyle{\rm ES}_p (X) = \frac {\theta}{p} \int_0^p \left[ Q^{-1} (\gamma, 1 - v) \right]^{1 / c} dv\end{array} [REMOVE_ME_2]$

for $x > 0$, $0 < p < 1$, $\theta > 0$, the scale parameter, $c > 0$, the first shape parameter, and $\gamma > 0$, the second shape parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for Stacy distribution due to Stacy (1962) given by

for $x > 0$, $0 < p < 1$, $\theta > 0$, the scale parameter, $c > 0$, the first shape parameter, and $\gamma > 0$, the second shape parameter.

dstacygamma(x, gamma=1, c=1, theta=1, log=FALSE)
pstacygamma(x, gamma=1, c=1, theta=1, log.p=FALSE, lower.tail=TRUE)
varstacygamma(p, gamma=1, c=1, theta=1, log.p=FALSE, lower.tail=TRUE)
esstacygamma(p, gamma=1, c=1, theta=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 scale parameter, must be positive, the default is 1
• c: the value of the first scale parameter, must be positive, the default is 1
• gamma: the value of the second scale 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)
dstacygamma(x)
pstacygamma(x)
varstacygamma(x)
esstacygamma(x)