kumnormal function

Kumaraswamy normal distribution

Computes the pdf, cdf, value at risk and expected shortfall for Kumaraswamy normal distribution due to Cordeiro and de Castro (2011) given by [REMOVE_ME]$\begin{array}{ll}&\displaystylef (x) = \frac {a b}{\sigma} \phi \left( \frac {x - \mu}{\sigma} \right) \Phi^{a - 1} \left( \frac {x - \mu}{\sigma} \right)\left[ 1 - \Phi^a \left( \frac {x - \mu}{\sigma} \right) \right]^{b - 1},\\&\displaystyleF (x) = 1 - \left[ 1 - \Phi^a \left( \frac {x - \mu}{\sigma} \right) \right]^b,\\&\displaystyle{\rm VaR}_p (X) = \mu + \sigma \Phi^{-1} \left( \left[ 1 - (1 - p)^{1 / b} \right]^{1 / a} \right),\\&\displaystyle{\rm ES}_p (X) = \mu + \frac {\sigma}{p} \int_0^p \Phi^{-1} \left( \left[ 1 - (1 - v)^{1 / b} \right]^{1 / a} \right) dv\end{array} [REMOVE_ME_2]$

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \mu < \infty$, the location parameter, $\sigma > 0$, the scale parameter, $a > 0$, the first shape parameter, and $b > 0$, the second shape parameter.

Description

Computes the pdf, cdf, value at risk and expected shortfall for Kumaraswamy normal distribution due to Cordeiro and de Castro (2011) given by

for $-\infty < x < \infty$, $0 < p < 1$, $-\infty < \mu < \infty$, the location parameter, $\sigma > 0$, the scale parameter, $a > 0$, the first shape parameter, and $b > 0$, the second shape parameter.

dkumnormal(x, mu=0, sigma=1, a=1, b=1, log=FALSE)
pkumnormal(x, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
varkumnormal(p, mu=0, sigma=1, a=1, b=1, log.p=FALSE, lower.tail=TRUE)
eskumnormal(p, mu=0, sigma=1, a=1, b=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
• mu: the value of the location parameter, can take any real value, the default is zero
• 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
• 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)
dkumnormal(x)
pkumnormal(x)
varkumnormal(x)
eskumnormal(x)