# Stacy distribution

### Description

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

\begin{array}{ll} &\displaystyle f (x) = \frac {c x^{c γ - 1} \exp ≤ft[ -(x / θ)^c \right]}{θ^{c γ} Γ (γ)}, \\ &\displaystyle F (x) = 1 - Q ≤ft( γ, ≤ft( \frac {x}{θ} \right)^c \right), \\ &\displaystyle {\rm VaR}_p (X) = θ ≤ft[ Q^{-1} (γ, 1 - p) \right]^{1 / c}, \\ &\displaystyle {\rm ES}_p (X) = \frac {θ}{p} \int_0^p ≤ft[ Q^{-1} (γ, 1 - v) \right]^{1 / c} dv \end{array}

for x > 0, 0 < p < 1, θ > 0, the scale parameter, c > 0, the first shape parameter, and γ > 0, the second shape parameter.

### Usage

 1 2 3 4 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

### 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

S. Nadarajah, S. Chan and E. Afuecheta, An R Package for value at risk and expected shortfall, submitted

### Examples

 1 2 3 4 5 x=runif(10,min=0,max=1) dstacygamma(x) pstacygamma(x) varstacygamma(x) esstacygamma(x) 

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