Box–Cox Cole and Green distribution (BCCG)
bccg.RdDensity, 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
Value
dbccg gives the density, pbccg gives the distribution function, qbccg gives the quantile function, and rbccg generates random deviates.
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.
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/.