asypower function

Asymmetric power distribution

Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric power distribution due to Komunjer (2007) given by [REMOVE_ME] \begin{array}{ll}&\displaystylef(x) = \left\{ \begin{array}{ll}\displaystyle\frac {\displaystyle \delta^{1 / \lambda}}{\displaystyle \Gamma (1 + 1 / \lambda)}\exp \left[ -\frac {\delta}{a^\lambda} |x|^\lambda \right], & \mbox{if $x \leq 0$},\\\\\displaystyle\frac {\displaystyle \delta^{1 / \lambda}}{\displaystyle \Gamma (1 + 1 / \lambda)}\exp \left[ -\frac {\delta}{(1 - a)^\lambda} |x|^\lambda \right], & \mbox{if $x > 0$,}\end{array}\right.\\&\displaystyleF (x) = \left\{ \begin{array}{ll}\displaystylea - a {\cal I} \left( \frac {\delta}{a^\lambda} \sqrt{\lambda} |x|^\lambda, 1 / \lambda \right), & \mbox{if $x \leq 0$,}\\\\\displaystylea - (1 - a) {\cal I} \left( \frac {\delta}{(1 - a)^\lambda}\sqrt{\lambda} |x|^\lambda, 1 / \lambda \right), & \mbox{if $x > 0$,}\end{array}\right.\\&\displaystyle{\rm VaR}_p (X) = \left\{ \begin{array}{ll}\displaystyle-\left[ \frac {a^\lambda}{\delta \sqrt{\lambda}} \right]^{1 / \lambda}\left[ {\cal I}^{-1} \left( 1 - \frac {p}{a}, \frac {1}{\lambda} \right) \right]^{1 / \lambda}, & \mbox{if $p \leq a$,}\\\\\displaystyle-\left[ \frac {(1 - a)^\lambda}{\delta \sqrt{\lambda}} \right]^{1 / \lambda}\left[ {\cal I}^{-1} \left( 1 - \frac {1 - p}{1 - a}, \frac {1}{\lambda} \right) \right]^{1 / \lambda}, & \mbox{if $p > a$,}\end{array}\right.\\&\displaystyle{\rm ES}_p (X) = \left\{ \begin{array}{ll}\displaystyle-\frac {1}{p} \left[ \frac {a^\lambda}{\delta \sqrt{\lambda}} \right]^{1 / \lambda}\int_0^p \left[ {\cal I}^{-1} \left( 1 - \frac {v}{a}, \frac {1}{\lambda} \right) \right]^{1 / \lambda} dv, &\mbox{if $p \leq a$,}\\\\\displaystyle-\frac {1}{p} \left[ \frac {a^\lambda}{\delta \sqrt{\lambda}} \right]^{1 / \lambda}\int_0^a \left[ {\cal I}^{-1} \left( 1 - \frac {v}{a}, \frac {1}{\lambda} \right) \right]^{1 / \lambda} dv\\\quad\displaystyle-\frac {1}{p} \left[ \frac {(1 - a)^\lambda}{\delta \sqrt{\lambda}} \right]^{1 / \lambda}\int_a^p \left[ {\cal I}^{-1} \left( 1 -\frac {1 - v}{1 - a}, \frac {1}{\lambda} \right) \right]^{1 / \lambda} dv, & \mbox{if $p > a$}\end{array}\right.\end{array} [REMOVE_ME_2]

for $-\infty < x < \infty$, $0 < p < 1$, $0 < a < 1$, the first scale parameter, $\delta > 0$, the second scale parameter, and $\lambda > 0$, the shape parameter, where ${\cal I} (x, \gamma) = \frac {1}{\Gamma (\gamma)} \int_0^{x \sqrt{\gamma}} t^{\gamma - 1} \exp (-t) dt$.

Description

Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric power distribution due to Komunjer (2007) given by

for $-\infty < x < \infty$, $0 < p < 1$, $0 < a < 1$, the first scale parameter, $\delta > 0$, the second scale parameter, and $\lambda > 0$, the shape parameter, where ${\cal I} (x, \gamma) = \frac {1}{\Gamma (\gamma)} \int_0^{x \sqrt{\gamma}} t^{\gamma - 1} \exp (-t) dt$.

dasypower(x, a=0.5, lambda=1, delta=1, log=FALSE)
pasypower(x, a=0.5, lambda=1, delta=1, log.p=FALSE, lower.tail=TRUE)
varasypower(p, a=0.5, lambda=1, delta=1, log.p=FALSE, lower.tail=TRUE)
esasypower(p, a=0.5, lambda=1, delta=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 in the unit interval, the default is 0.5
• delta: the value of the second scale parameter, must be positive, the default is 1
• lambda: 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

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)
dasypower(x)
pasypower(x)
varasypower(x)
esasypower(x)