xie {VaRES}R Documentation

Xie distribution

Description

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

f(x)=λb(xa)b1exp[(x/a)b]exp(λa)exp{λaexp[(x/a)b]},F(x)=1exp(λa)exp{λaexp[(x/a)b]},VaRp(X)=a{log[1log(1p)λa]}1/b,ESp(X)=ap0p{log[1log(1v)λa]}1/bdv\begin{array}{ll} &\displaystyle f(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\}, \\ &\displaystyle F (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}

for x>0x > 0, 0<p<10 < p < 1, a>0a > 0, the first scale parameter, b>0b > 0, the shape parameter, and λ>0\lambda > 0, the second scale parameter.

Usage

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

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

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, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
dxie(x)
pxie(x)
varxie(x)
esxie(x)

[Package VaRES version 1.0.2 Index]