expweibull function

Exponentiated Weibull distribution

Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated Weibull distribution due to Mudholkar and Srivastava (1993) and Mudholkar et al. (1995) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef(x) = a \alpha \sigma^{-\alpha} x^{\alpha - 1}\exp \left[ -(x / \sigma)^\alpha \right]\left\{ 1 - \exp \left[ -(x / \sigma)^\alpha \right] \right\}^{a - 1},\\&\displaystyleF (x) = \left\{ 1 - \exp \left[ -(x / \sigma)^\alpha \right] \right\}^a,\\&\displaystyle{\rm VaR}_p (X) = \sigma \left[ -\log \left( 1 - p^{1 / a} \right) \right]^{1 / \alpha},\\&\displaystyle{\rm ES}_p (X) = \frac {\sigma}{p} \int_0^p \left[ -\log \left( 1 - v^{1 / a} \right) \right]^{1 / \alpha} dv\end{array} [REMOVE_ME_2]$

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

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponentiated Weibull distribution due to Mudholkar and Srivastava (1993) and Mudholkar et al. (1995) given by

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

dexpweibull(x, a=1, alpha=1, sigma=1, log=FALSE)
pexpweibull(x, a=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varexpweibull(p, a=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
esexpweibull(p, a=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
• alpha: the value of the second 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)
dexpweibull(x)
pexpweibull(x)
varexpweibull(x)
esexpweibull(x)