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)mu.link, with "identity" link as the default for the mu parameter. Other links are "inverse", "log" and "own"sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse","identity", "own"nu.link, with "identity" link as the default for the nu parameter. Other links are "inverse", "log", "own"tau.link, with "log" link as the default for the tau parameter. Other links are "inverse", "identity" and "own"nu parameter valuestau parameter valueslength(n) > 1, the length is
taken to be the number requiredBCT() 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.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. $\mu>0,\sigma>0, \nu=(-\infty,\infty) {\rm and} \tau>0$ ]
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.
Stasinopoulos, D. M. Rigby, R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R.
Accompanying documentation in the current GAMLSS help files, (see also
gamlss.family, BCPE, BCCGBCT() # 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")Run the code above in your browser using DataLab