kumburr7 function

Kumaraswamy Burr XII distribution

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Burr XII distribution due to Parana'iba et al. (2013) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {a b k c x^{c - 1}}{\left( 1 + x^c \right)^{k + 1}} \left[ 1 - \left( 1 + x^c \right)^{-k} \right]^{a - 1}\left\{ 1 - \left[ 1 - \left( 1 + x^c \right)^{-k} \right]^a \right\}^{b - 1},\\&\displaystyleF (x) = 1 - \left\{ 1 - \left[ 1 - \left( 1 + x^c \right)^{-k} \right]^a \right\}^b,\\&\displaystyle{\rm VaR}_p (X) = \left[ \left\{ 1 - \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} \right\}^{-1 / k} - 1 \right]^{1/c},\\&\displaystyle{\rm ES}_p (X) = \frac {1}{p} \int_0^p \left[ \left\{ 1 - \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} \right\}^{-1 / k} - 1 \right]^{1/c} dv\end{array} [REMOVE_ME_2]$

for $x > 0$, $0 < p < 1$, $a > 0$, the first shape parameter, $b > 0$, the second shape parameter, $c > 0$, the third shape parameter, and $k > 0$, the fourth shape parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the Kumaraswamy Burr XII distribution due to Parana'iba et al. (2013) given by

for $x > 0$, $0 < p < 1$, $a > 0$, the first shape parameter, $b > 0$, the second shape parameter, $c > 0$, the third shape parameter, and $k > 0$, the fourth shape parameter.

dkumburr7(x, a=1, b=1, k=1, c=1, log=FALSE)
pkumburr7(x, a=1, b=1, k=1, c=1, log.p=FALSE, lower.tail=TRUE)
varkumburr7(p, a=1, b=1, k=1, c=1, log.p=FALSE, lower.tail=TRUE)
eskumburr7(p, a=1, b=1, k=1, c=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 shape parameter, must be positive, the default is 1
• b: the value of the second shape parameter, must be positive, the default is 1
• c: the value of the third shape parameter, must be positive, the default is 1
• k: the value of the fourth 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)
dkumburr7(x)
pkumburr7(x)
varkumburr7(x)
eskumburr7(x)