bccg: Box–Cox Cole and Green distribution (BCCG)

In RTMBdist: Distributions Compatible with Automatic Differentiation by 'RTMB'

Box–Cox Cole and Green distribution (BCCG)

Description

Density, distribution function, quantile function, and random generation for the Box–Cox Cole and Green distribution.

Usage

dbccg(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)

pbccg(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)

qbccg(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)

rbccg(n, mu = 1, sigma = 0.1, nu = 1)

Arguments

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

Details

dbccg and pbccg allow for automatic differentiation with RTMB. The parameterisation follows the BCCG family of the gamlss.dist package.

The density is

f(x; \mu, \sigma, \nu) = \frac{x^{\nu-1}}{\mu^{\nu} \sigma \sqrt{2\pi}} \exp\!\left(-\tfrac{z^2}{2}\right) \Bigl[\Phi\!\left(\tfrac{1}{\sigma|\nu|}\right)\Bigr]^{-1}, \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 \Phi is the standard normal CDF.

Value

dbccg gives the density, pbccg gives the distribution function, qbccg gives the quantile function, and rbccg generates random deviates.

References

Cole, T. J. and Green, P. J. (1992) Smoothing reference centile curves: the LMS method and penalized likelihood. Statistics in Medicine, 11, 1305-1319.

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/.

See Also

bct, bcpe, gengamma

Examples

x <- rbccg(5, mu = 10, sigma = 0.2, nu = 0.5)
d <- dbccg(x, mu = 10, sigma = 0.2, nu = 0.5)
p <- pbccg(x, mu = 10, sigma = 0.2, nu = 0.5)
q <- qbccg(p, mu = 10, sigma = 0.2, nu = 0.5)

RTMBdist documentation built on Oct. 31, 2026, 5:07 p.m.