Description Usage Arguments Details Value Warning Note Author(s) References See Also Examples

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 | ```
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` |
Defines the |

`sigma.link` |
Defines the |

`nu.link` |
Defines the |

`tau.link` |
Defines the |

`x,q` |
vector of quantiles |

`mu` |
vector of location parameter values |

`sigma` |
vector of scale parameter values |

`nu` |
vector of |

`tau` |
vector of |

`log, log.p` |
logical; if TRUE, probabilities p are given as log(p). |

`lower.tail` |
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x] |

`p` |
vector of probabilities. |

`n` |
number of observations. If |

The probability density function of the untruncated Box-Cox t distribution, `BCTuntr`

, is given by

*f(y|mu,sigma,nu,tau)=(1/(y*sigma))*(Γ((tau+1)/2)/(Gamma(1/2)*Gamma(tau/2)*tau^0.5))*(1+z^2/tau)^(-(tau+1)/2)*

where if *ν!=0* then *z=[(y/mu)^(nu)-1]/(nu*sigma)* else *z=log(y/μ)/σ*,
for *y>0*, *μ>0*, *σ>0*, *ν=(-Inf,+Inf)* and *τ>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.

`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=(-Inf,+Inf) and tau>0* ]

*mu* is the median of the distribution, *sigma*(tau/(tau-2))^0.5*
is approximate the coefficient of variation (for small *sigma* and moderate `nu>0`

and moderate or large *tau*),
*nu* controls the skewness and *tau* the kurtosis of the distribution

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

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.

1 2 3 4 5 6 7 8 9 | ```
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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