Power Exponential distribution (PE and PE2)

Density, distribution function, quantile function, and random generation for the Power Exponential distribution (two versions).

Usage

dpowerexp(x, mu = 0, sigma = 1, nu = 2, log = FALSE)

ppowerexp(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

qpowerexp(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

rpowerexp(n, mu = 0, sigma = 1, nu = 2)

dpowerexp2(x, mu = 0, sigma = 1, nu = 2, log = FALSE)

ppowerexp2(q, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

qpowerexp2(p, mu = 0, sigma = 1, nu = 2, lower.tail = TRUE, log.p = FALSE)

rpowerexp2(n, mu = 0, sigma = 1, nu = 2)

Arguments

x, q: vector of quantiles
mu: location parameter
sigma: scale parameter, must be positive
nu: shape parameter (real)
log, log.p: logical; if TRUE, probabilities/ densities $p$ are returned as $\log(p)$
lower.tail: logical; if TRUE (default), probabilities are $P[X \le x]$, otherwise $P[X > x]$
p: vector of probabilities
n: number of random values to return

Value

dpowerexp gives the density, ppowerexp gives the distribution function, qpowerexp gives the quantile function, and rpowerexp generates random deviates.

Details

The densities and distribution functions allow for automatic differentiation with RTMB.

For powerexp, mu is the mean and sigma is the standard deviation while this does not hold for powerexp2.

The parameterisation follows the PE and PE2 families of the gamlss.dist package.

For powerexp (PE), $\sigma$ is the standard deviation; the density is $$f(x;\,\mu,\sigma,\nu) = \frac{\nu}{2c\,\sigma\,\Gamma(1/\nu)} \exp\!\left(-\tfrac{1}{2}\left|\frac{x-\mu}{c\sigma}\right|^\nu\right),$$ where $c = [2^{-2/\nu}\Gamma(1/\nu)/\Gamma(3/\nu)]^{1/2}$.

For powerexp2 (PE2), $\sigma$ is a scale parameter; the density is $$f(x;\,\mu,\sigma,\nu) = \frac{\nu}{2\sigma\,\Gamma(1/\nu)} \exp\!\left(-\left|\frac{x-\mu}{\sigma}\right|^\nu\right).$$

References

Nelson, D. B. (1991) Conditional heteroskedasticity in asset returns: a new approach. Econometrica, 59, 347-370.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

Examples

# PE
x <- rpowerexp(1, mu = 0, sigma = 1, nu = 2)
d <- dpowerexp(x, mu = 0, sigma = 1, nu = 2)
p <- ppowerexp(x, mu = 0, sigma = 1, nu = 2)
q <- qpowerexp(p, mu = 0, sigma = 1, nu = 2)

# PE2
x <- rpowerexp2(1, mu = 0, sigma = 1, nu = 2)
d <- dpowerexp2(x, mu = 0, sigma = 1, nu = 2)
p <- ppowerexp2(x, mu = 0, sigma = 1, nu = 2)
q <- qpowerexp2(p, mu = 0, sigma = 1, nu = 2)