bct: Box–Cox t distribution (BCT)

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

Box–Cox t distribution (BCT)

Description

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

Usage

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

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

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

rbct(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	skewness parameter (real).
tau	degrees of freedom, 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

Details

dbct and pbct allow for automatic differentiation with RTMB. The parameterisation follows the BCT 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 Student's t distribution with \tau degrees of freedom.

Value

dbct gives the density, pbct gives the distribution function, qbct gives the quantile function, and rbct generates random deviates.

References

Rigby, R. A. and Stasinopoulos, D. M. (2006) Using the Box-Cox t distribution in GAMLSS to model skewness and kurtosis. Statistical Modelling, 6(3), 209. doi:10.1191/1471082X06st122oa

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

bccg, bcpe, skewt

Examples

x <- rbct(1, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)
d <- dbct(x, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)
p <- pbct(x, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)
q <- qbct(p, mu = 10, sigma = 0.2, nu = 0.5, tau = 4)

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