Box-Cox Power Exponential distribution (BCPE)

Density, distribution function, quantile function, and random generation for the Box-Cox Power Exponential distribution.

Usage

dbcpe(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)

pbcpe(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)

qbcpe(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)

rbcpe(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)

Arguments

x, q: vector of quantiles
mu: location parameter, must be positive.
sigma: scale parameter, must be positive.
nu: vector of nu parameter values.
tau: vector of tau parameter values, must be positive.
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

dbcpe gives the density, pbcpe gives the distribution function, qbcpe gives the quantile function, and rbcpe generates random deviates.

Details

dbcpe and pbcpe allow for automatic differentiation with RTMB. The parameterisation follows the BCPE family of the gamlss.dist package.

The density is $$f(x; \mu, \sigma, \nu, \tau) = \frac{x^{\nu-1}}{\mu^{\nu} \sigma} \frac{f_T(z;\tau)}{F_T\!\left(1/(\sigma|\nu|);\tau\right)}, \quad x > 0,$$ where $z = [(x/\mu)^\nu - 1]/(\nu\sigma)$ for $\nu \neq 0$ and $z = \log(x/\mu)/\sigma$ for $\nu = 0$, and $f_T(\cdot;\tau)$ and $F_T(\cdot;\tau)$ are the PDF and CDF of the power exponential (PE) distribution with shape $\tau$.

References

Rigby, R. A. and Stasinopoulos, D. M. (2004) Smooth centile curves for skew and kurtotic data modelled using the Box-Cox Power Exponential distribution. Statistics in Medicine, 23, 3053-3076.

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

x <- rbcpe(1, mu = 5, sigma = 0.1, nu = 1, tau = 1)
d <- dbcpe(x, mu = 5, sigma = 0.1, nu = 1, tau = 1)
p <- pbcpe(x, mu = 5, sigma = 0.1, nu = 1, tau = 1)
q <- qbcpe(p, mu = 5, sigma = 0.1, nu = 1, tau = 1)