The function BCT() defines the Box-Cox t distribution, a four parameter distribution,
for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The functions dBCT,
pBCT, qBCT and rBCT define the density, distribution function, quantile function and random
generation for the Box-Cox t distribution.
[The function BCTuntr() is the original version of the function suitable only for the untruncated BCT distribution].
See Rigby and Stasinopoulos (2003) for details.
The function BCT is identical to BCT but with log link for mu.
BCT(mu.link = "identity", sigma.link = "log", nu.link = "identity",
tau.link = "log")
BCTo(mu.link = "log", sigma.link = "log", nu.link = "identity",
tau.link = "log")
BCTuntr(mu.link = "identity", sigma.link = "log", nu.link = "identity",
tau.link = "log")
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)
dBCTo(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCTo(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
qBCTo(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
rBCTo(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)Defines the mu.link, with "identity" link as the default for the mu parameter. Other links are "inverse", "log" and "own"
Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse","identity", "own"
Defines the nu.link, with "identity" link as the default for the nu parameter. Other links are "inverse", "log", "own"
Defines the tau.link, with "log" link as the default for the tau parameter. Other links are "inverse", "identity" and "own"
vector of quantiles
vector of location parameter values
vector of scale parameter values
vector of nu parameter values
vector of tau parameter values
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
vector of probabilities.
number of observations. If length(n) > 1, the length is
taken to be the number required
BCT() returns a gamlss.family object which can be used to fit a Box Cox-t distribution in the gamlss() function.
dBCT() gives the density, pBCT() gives the distribution
function, qBCT() gives the quantile function, and rBCT()
generates random deviates.
The use BCTuntr distribution may be unsuitable for some combinations of the parameters (mainly for large \(\sigma\))
where the integrating constant is less than 0.99. A warning will be given if this is the case.
The BCT distribution is suitable for all combinations of the parameters within their ranges
[i.e. \(\mu>0,\sigma>0, \nu=(-\infty,\infty) {\rm and} \tau>0\) ]
The probability density function of the untruncated Box-Cox t distribution, BCTuntr, is given by
$$f(y|\mu,\sigma,\nu,\tau)=\frac{y^{\nu-1}}{\mu^{\nu}\sigma} \frac{\Gamma[(\tau+1)/2]}{\Gamma(1/2) \Gamma(\tau/2) \tau^{0.5}} [1+(1/\tau)z^2]^{-(\tau+1)/2}$$
where if \(\nu \neq 0\) then \(z=[(y/\mu)^{\nu}-1]/(\nu \sigma)\) else \(z=\log(y/\mu)/\sigma\),
for \(y>0\), \(\mu>0\), \(\sigma>0\), \(\nu=(-\infty,+\infty)\) and \(\tau>0\).
The Box-Cox t distribution, BCT, adjusts the above density \(f(y|\mu,\sigma,\nu,\tau)\) for the
truncation resulting from the condition \(y>0\). See Rigby and Stasinopoulos (2003) for details.
Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.
Rigby, R.A. Stasinopoulos, D.M. (2006). Using the Box-Cox t distribution in GAMLSS to mode skewnees and and kurtosis. to appear in Statistical Modelling.
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. An older version can be found in http://www.gamlss.com/.
Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, http://www.jstatsoft.org/v23/i07.
Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC.
# NOT RUN {
BCT() # gives information about the default links for the Box Cox t distribution
# library(gamlss)
#data(abdom)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=BCT, data=abdom) #
#plot(h)
plot(function(x)dBCT(x, mu=5,sigma=.5,nu=1, tau=2), 0.0, 20,
main = "The BCT density mu=5,sigma=.5,nu=1, tau=2")
plot(function(x) pBCT(x, mu=5,sigma=.5,nu=1, tau=2), 0.0, 20,
main = "The BCT cdf mu=5, sigma=.5, nu=1, tau=2")
# }
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