xie function

Xie distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Xie distribution due to Xie et al. (2002) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef(x) = \lambda b \left( \frac {x}{a} \right)^{b - 1}\exp \left[ (x/a)^b \right] \exp \left( \lambda a \right)\exp \left\{ -\lambda a \exp \left[ (x/a)^b \right] \right\},\\&\displaystyleF (x) =1 - \exp \left( \lambda a \right)\exp \left\{ -\lambda a \exp \left[ (x/a)^b \right] \right\},\\&\displaystyle{\rm VaR}_p (X) =a \left\{ \log \left[ 1 - \frac {\log (1 - p)}{\lambda a} \right] \right\}^{1/b},\\&\displaystyle{\rm ES}_p (X) =\frac {a}{p} \int_0^p \left\{ \log \left[ 1 - \frac {\log (1 - v)}{\lambda a} \right] \right\}^{1/b} dv\end{array} [REMOVE_ME_2]$

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

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Xie distribution due to Xie et al. (2002) given by

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

dxie(x, a=1, b=1, lambda=1, log=FALSE)
pxie(x, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
varxie(p, a=1, b=1, lambda=1, log.p=FALSE, lower.tail=TRUE)
esxie(p, a=1, b=1, lambda=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
• a: the value of the first scale parameter, must be positive, the default is 1
• lambda: the value of the second scale parameter, must be positive, the default is 1
• b: 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)
dxie(x)
pxie(x)
varxie(x)
esxie(x)