Kumaraswamy Weibull distribution
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Weibull distribution due to Cordeiro et al. (2010) given by [REMOVE_ME]\displaystylef(x)=σαabαxα−1exp[−(σx)α]{1−exp[−(σx)α]}a−1[1−{1−exp[−(σx)α]}a]b−1,\displaystyleF(x)=1−[1−{1−exp[−(σx)α]}a]b,VaRp(X)=σ[−log{1−[1−(1−p)1/b]1/a}]1/α,ESp(X)=pσ∫0p[−log{1−[1−(1−v)1/b]1/a}]1/αdv[REMOVEME2]
for x>0, 0<p<1, a>0, the first shape parameter, b>0, the second shape parameter, α>0, the third shape parameter, and σ>0, the scale parameter.
Description
Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Weibull distribution due to Cordeiro et al. (2010) given by
\displaystylef(x)=σαabαxα−1exp[−(σx)α]{1−exp[−(σx)α]}a−1[1−{1−exp[−(σx)α]}a]b−1,\displaystyleF(x)=1−[1−{1−exp[−(σx)α]}a]b,VaRp(X)=σ[−log{1−[1−(1−p)1/b]1/a}]1/α,ESp(X)=pσ∫0p[−log{1−[1−(1−v)1/b]1/a}]1/αdv
for x>0, 0<p<1, a>0, the first shape parameter, b>0, the second shape parameter, α>0, the third shape parameter, and σ>0, the scale parameter.
dkumweibull(x, a=1, b=1, alpha=1, sigma=1, log=FALSE)
pkumweibull(x, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
varkumweibull(p, a=1, b=1, alpha=1, sigma=1, log.p=FALSE, lower.tail=TRUE)
eskumweibull(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)
dkumweibull(x)
pkumweibull(x)
varkumweibull(x)
eskumweibull(x)