aep {VaRES}

Asymmetric exponential power distribution

Description

Computes the pdf, cdf, value at risk and expected shortfall for the asymmetric exponential power distribution due to Zhu and Zinde-Walsh (2009) given by

\begin{array}{ll} &\displaystyle f (x) = \left\{ \begin{array}{ll} \displaystyle \frac {\alpha}{\alpha^{*}} K \left( q_1 \right) \exp \left[ -\frac {1}{q_1} \left | \frac {x}{2 \alpha^{*}} \right |^{q_1} \right], & \mbox{if $x \leq 0$,} \\ \\ \displaystyle \frac {1 - \alpha}{1 - \alpha^{*}} K \left( q_2 \right) \exp \left[ -\frac {1}{q_2} \left | \frac {x}{2 - 2 \alpha^{*}} \right |^{q_2} \right], & \mbox{if $x > 0$,} \end{array} \right. \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle \alpha Q \left( \frac {1}{q_1} \left( \frac {\mid x \mid}{2 \alpha^{*}} \right)^{q_1}, \frac {1}{q_1} \right), & \mbox{if $x \leq 0$,} \\ \\ \displaystyle 1 - (1 - \alpha) Q \left( \frac {1}{q_2} \left( \frac {\mid x \mid}{2 - 2 \alpha^{*}} \right)^{q_2}, \frac {1}{q_2} \right), & \mbox{if $x > 0$,} \end{array} \right. \\ &\displaystyle {\rm VaR}_p (X) = \left\{ \begin{array}{ll} \displaystyle -2 \alpha^{*} \left[ q_1 Q^{-1} \left( \frac {p}{\alpha}, \frac {1}{q_1} \right) \right]^{\frac {1}{q_1}}, & \mbox{if $p \leq \alpha$,} \\ \\ \displaystyle 2 \left(1 - \alpha^{*}\right) \left[ q_2 Q^{-1} \left( \frac {1 - p}{1 - \alpha}, \frac {1}{q_2} \right) \right]^{\frac {1}{q_2}}, & \mbox{if $p > \alpha$,} \end{array} \right. \\ &\displaystyle {\rm ES}_p (X) = \left\{ \begin{array}{ll} \displaystyle -\frac {2 \alpha^{*}}{p} \int_0^p \left[ q_1 Q^{-1} \left( \frac {v}{\alpha}, \frac {1}{q_1} \right) \right]^{\frac {1}{q_1}} dv, & \mbox{if $p \leq \alpha$,} \\ \\ \displaystyle -\frac {2 \alpha^{*}}{p} \int_0^\alpha \left[ q_1 Q^{-1} \left( \frac {v}{\alpha}, \frac {1}{q_1} \right) \right]^{\frac {1}{q_1}} dv & \ \\ \quad \displaystyle +\frac {2 \left(1 - \alpha^{*}\right)}{p} \int_\alpha^p \left[ q_2 Q^{-1} \left( \frac {1 - v}{1 - \alpha}, \frac {1}{q_2} \right) \right]^{\frac {1}{q_2}} dv, & \mbox{if $p > \alpha$} \end{array} \right. \end{array}

for -\infty < x < \infty, 0 < p < 1, 0 < \alpha < 1, the scale parameter, q_1 > 0, the first shape parameter, and q_2 > 0, the second shape parameter, where \alpha^{*} = \alpha K \left( q_1 \right) / \left\{ \alpha K \left( q_1 \right) + (1 - \alpha) K \left( q_2 \right) \right\}, K (q) = \frac {1}{2 q^{1/q} \Gamma (1 + 1/q)}, Q (a, x) = \int_x^\infty t^{a - 1} \exp \left( -t \right) dt / \Gamma (a) denotes the regularized complementary incomplete gamma function, \Gamma (a) = \int_0^\infty t^{a - 1} \exp \left( -t \right) dt denotes the gamma function, and Q^{-1} (a, x) denotes the inverse of Q (a, x).

Usage

daep(x, q1=1, q2=1, alpha=0.5, log=FALSE)
paep(x, q1=1, q2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)
varaep(p, q1=1, q2=1, alpha=0.5, log.p=FALSE, lower.tail=TRUE)
esaep(p, q1=1, q2=1, alpha=0.5)

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 in the unit interval, the default is 0.5

q1

the value of the first shape parameter, must be positive, the default is 1

q2

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

Value

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.

Author(s)

Saralees Nadarajah

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, doi:10.1080/03610918.2014.944658

Examples

x=runif(10,min=0,max=1)
daep(x)
paep(x)
varaep(x)
esaep(x)