gamlss.dist (version 6.1-11)

BCT: Box-Cox t distribution for fitting a GAMLSS

Description

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 BCTo is identical to BCT but with log link for mu.

Usage

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)

Value

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.

Details

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 \(\tau>0\) see pp. 450-451 of Rigby et al. (2019).

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.

References

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. Statistical Modelling 6(3):200. tools:::Rd_expr_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, tools:::Rd_expr_doi("10.1201/9780429298547"). An older version can be found in https://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, tools:::Rd_expr_doi("10.18637/jss.v023.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. tools:::Rd_expr_doi("10.1201/b21973")

(see also https://www.gamlss.com/).

See Also

gamlss.family, BCPE, BCCG

Examples

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")