aep function

Asymmetric exponential power distribution

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 [REMOVE_ME] \begin{array}{ll}&\displaystylef (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.\\&\displaystyleF (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 $x \leq 0$,}\\\\\displaystyle1 - (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 $p \leq \alpha$,}\\\\\displaystyle2 \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} [REMOVE_ME_2]

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 $Q (a, x)$.

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

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 $Q (a, x)$.

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

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)
daep(x)
paep(x)
varaep(x)
esaep(x)