BCT | R Documentation |

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.

```
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)=\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.

`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`

]

`\mu`

is the median of the distribution, `\sigma(\frac{\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. *Statistical Modelling* 6(3):200. \Sexpr[results=rd]{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, \Sexpr[results=rd]{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, \Sexpr[results=rd]{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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}

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

`gamlss.family`

, `BCPE`

, `BCCG`

```
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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