kumloglogis function

Kumaraswamy log-logistic distribution

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

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

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy log-logistic distribution due to de Santana et al. (2012) given by

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

dkumloglogis(x, a=1, b=1, alpha=1, beta=1, log=FALSE)
pkumloglogis(x, a=1, b=1, alpha=1, beta=1, log.p=FALSE, lower.tail=TRUE)
varkumloglogis(p, a=1, b=1, alpha=1, beta=1, log.p=FALSE, lower.tail=TRUE)
eskumloglogis(p, a=1, b=1, alpha=1, beta=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
• alpha: the value of the scale parameter, must be positive, the default is 1
• beta: the value of the first shape parameter, must be positive, the default is 1
• a: the value of the second shape parameter, must be positive, the default is 1
• b: 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)
dkumloglogis(x)
pkumloglogis(x)
varkumloglogis(x)
eskumloglogis(x)