exppower: Exponential power distribution

In VaRES: Computes Value at Risk and Expected Shortfall for over 100 Parametric Distributions

Exponential power distribution

Description

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

\begin{array}{ll} &\displaystyle f (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\}, \\ &\displaystyle F (x) = \left\{ \begin{array}{ll} \displaystyle \frac {1}{2} Q \left( \frac {1}{a}, \frac {(\mu - x)^a}{a \sigma^a} \right), & \mbox{if $x \leq \mu$,} \\ \\ \displaystyle 1 - \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}

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

Usage

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

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, \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/03610918.2014.944658")}

Examples

x=runif(10,min=0,max=1)
dexppower(x)
pexppower(x)
varexppower(x)
esexppower(x)

VaRES documentation built on April 22, 2023, 1:16 a.m.