exppower function

Exponential power distribution

Computes the pdf, cdf, value at risk and expected shortfall for the exponential power distribution due to Subbotin (1923) given by [REMOVE_ME] \begin{array}{ll}&\displaystylef (x) = \frac {1}{\displaystyle 2 a^{1/a} \sigma \Gamma \left( 1 + 1/a \right)}\exp \left\{ -\frac {\mid x - \mu \mid^a}{a \sigma^a} \right\},\\&\displaystyleF (x) =\left\{\begin{array}{ll}\displaystyle\frac {1}{2} Q \left( \frac {1}{a}, \frac {(\mu - x)^a}{a \sigma^a} \right), & \mbox{if $x \leq \mu$,}\\\\\displaystyle1 - \frac {1}{2} Q \left( \frac {1}{a}, \frac {(x - \mu)^a}{a \sigma^a} \right), & \mbox{if $x > \mu$,}\end{array}\right.\\&\displaystyle{\rm VaR}_p (X) =\left\{\begin{array}{ll}\displaystyle\mu - a^{1/a} \sigma \left[ Q^{-1} \left( \frac {1}{a}, 2 p \right) \right]^{1/a}, & \mbox{if $p \leq 1/2$,}\\\\\mu + a^{1/a} \sigma \left[ Q^{-1} \left( \frac {1}{a}, 2 (1 - p) \right) \right]^{1/a}, & \mbox{if $p > 1/2$,}\end{array}\right.\\&\displaystyle{\rm ES}_p (X) =\left\{\begin{array}{ll}\displaystyle\mu - \frac {a^{1/a} \sigma}{p} \int_0^p \left[ Q^{-1} \left( \frac {1}{a}, 2 v \right) \right]^{1/a} dv, & \mbox{if $p \leq 1/2$,}\\\\\displaystyle\mu - \frac {a^{1/a} \sigma}{p} \int_0^{1/2} \left[ Q^{-1} \left( \frac {1}{a}, 2 v \right) \right]^{1/a} dv\\\displaystyle\quad+\frac {a^{1/a} \sigma}{p} \int_{1/2}^p \left[ Q^{-1} \left( \frac {1}{a}, 2 (1 - v) \right) \right]^{1/a} dv, & \mbox{if $p > 1/2$}\end{array}\right.\end{array} [REMOVE_ME_2]

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

Description

Computes the pdf, cdf, value at risk and expected shortfall for the exponential power distribution due to Subbotin (1923) given by

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

dexppower(x, mu=0, sigma=1, a=1, log=FALSE)
pexppower(x, mu=0, sigma=1, a=1, log.p=FALSE, lower.tail=TRUE)
varexppower(p, mu=0, sigma=1, a=1, log.p=FALSE, lower.tail=TRUE)
esexppower(p, mu=0, sigma=1, a=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 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)
dexppower(x)
pexppower(x)
varexppower(x)
esexppower(x)