betaweibull function

Beta Weibull distribution

Computes the pdf, cdf, value at risk and expected shortfall for the beta Weibull distribution due to Cordeiro et al. (2012b) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef(x) = \frac {\alpha x^{\alpha - 1}}{\sigma^\alpha B (a, b)}\exp \left\{ -b \left( \frac {x}{\sigma} \right)^{\alpha} \right\}\left[ 1 - \exp \left\{ -\left( \frac {x}{\sigma} \right)^{\alpha} \right\} \right]^{a - 1},\\&\displaystyleF(x) = I_{1 - \exp \left\{ -\left( \frac {x}{\sigma} \right)^{\alpha} \right\}} (a, b),\\&\displaystyle{\rm VaR}_p (X) = \sigma \left\{ -\log \left[ 1 - I_p^{-1} (a, b) \right] \right\}^{1 / \alpha},\\&\displaystyle{\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \left\{ -\log \left[ 1 - I_v^{-1} (a, b) \right] \right\}^{1 / \alpha} dv\end{array} [REMOVE_ME_2]$

for $x > 0$, $0 < p < 1$, $a > 0$, the first shape parameter, $b > 0$, the second shape parameter, $\alpha > 0$, the third shape parameter, and $\sigma > 0$, the scale parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the beta Weibull distribution due to Cordeiro et al. (2012b) given by

for $x > 0$, $0 < p < 1$, $a > 0$, the first shape parameter, $b > 0$, the second shape parameter, $\alpha > 0$, the third shape parameter, and $\sigma > 0$, the scale parameter.

dbetaweibull(x, a=1, b=1, alpha=1, sigma=1, log=FALSE)
pbetaweibull(x, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varbetaweibull(p, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
esbetaweibull(p, a=1, b=1, alpha=1, sigma=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
• sigma: the value of the scale parameter, must be positive, the default is 1
• a: the value of the first shape parameter, must be positive, the default is 1
• b: the value of the second shape parameter, must be positive, the default is 1
• alpha: the value of the third 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)
dbetaweibull(x)
pbetaweibull(x)
varbetaweibull(x)
esbetaweibull(x)