# BCCG

##### Box-Cox Cole and Green distribution (or Box-Cox normal) for fitting a GAMLSS

The function `BCCG`

defines the Box-Cox Cole and Green distribution (Box-Cox normal), a three parameter distribution,
for a `gamlss.family`

object to be used in GAMLSS fitting using the function `gamlss()`

.
The functions `dBCCG`

, `pBCCG`

, `qBCCG`

and `rBCCG`

define the density, distribution function, quantile function and random generation for the specific parameterization of the Box-Cox Cole and Green distribution.
[The function `BCCGuntr()`

is the original version of the function suitable only for the untruncated Box-Cox Cole and Green distribution
See Cole and Green (1992) and Rigby and Stasinopoulos (2003a,2003b) for details.
The function `BCCGo`

is identical to `BCCG`

but with log link for mu.

- Keywords
- distribution, regression

##### Usage

```
BCCG(mu.link = "identity", sigma.link = "log", nu.link = "identity")
BCCGo(mu.link = "log", sigma.link = "log", nu.link = "identity")
BCCGuntr(mu.link = "identity", sigma.link = "log", nu.link = "identity")
dBCCG(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)
pBCCG(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qBCCG(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
rBCCG(n, mu = 1, sigma = 0.1, nu = 1)
dBCCGo(x, mu = 1, sigma = 0.1, nu = 1, log = FALSE)
pBCCGo(q, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
qBCCGo(p, mu = 1, sigma = 0.1, nu = 1, lower.tail = TRUE, log.p = FALSE)
rBCCGo(n, mu = 1, sigma = 0.1, nu = 1)
```

##### Arguments

- mu.link
Defines the

`mu.link`

, with "identity" link as the default for the mu parameter, other links are "inverse", "log" and "own"- sigma.link
Defines the

`sigma.link`

, with "log" link as the default for the sigma parameter, other links are "inverse", "identity" and "own"- nu.link
Defines the

`nu.link`

, with "identity" link as the default for the nu parameter, other links are "inverse", "log" and "own"- x,q
vector of quantiles

- mu
vector of location parameter values

- sigma
vector of scale parameter values

- nu
vector of skewness parameter values

- 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

`length(n) > 1`

, the length is taken to be the number required

##### Details

The probability distribution function of the untrucated Box-Cox Cole and Green distribution, `BCCGuntr`

, is defined as
$$f(y|\mu,\sigma,\nu)=\frac{1}{\sqrt{2\pi}\sigma}\frac{y^{\nu-1}}{\mu^\nu} \exp(-\frac{z^2}{2})$$

where if \(\nu \neq 0\) then \(z=[(y/\mu)^{\nu}-1]/(\nu \sigma)\) else \(z=\log(y/\mu)/\sigma\), for \(y>0\), \(\mu>0\), \(\sigma>0\) and \(\nu=(-\infty,+\infty)\).

The Box-Cox Cole anf Green distribution, `BCCG`

, adjusts the above density \(f(y|\mu,\sigma,\nu)\) for the
truncation resulting from the condition \(y>0\). See Rigby and Stasinopoulos (2003a,2003b) for details.

##### Value

`BCCG()`

returns a `gamlss.family`

object which can be used to fit a Cole and Green distribution in the `gamlss()`

function.
`dBCCG()`

gives the density, `pBCCG()`

gives the distribution
function, `qBCCG()`

gives the quantile function, and `rBCCG()`

generates random deviates.

##### Note

\(\mu\) is the median of the distribution \(\sigma\) is approximately the coefficient of variation (for small values of \(\sigma\)), and \(\nu\) controls the skewness.

The `BCCG`

distribution is suitable for all combinations of the parameters within their ranges
[i.e. \(\mu>0,\sigma>0, {\rm and} \nu=(-\infty,\infty)\) ]

##### Warning

The `BCCGuntr`

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 BCCG distribution is suitable for all combinations of the distributional parameters within
their range [i.e. \(\mu>0\), \(\sigma>0\), \(\nu=(-\infty, +\infty)\)]

##### References

Cole, T. J. and Green, P. J. (1992) Smoothing reference centile curves: the LMS method and penalized likelihood, *Statist. Med.* **11**, 1305--1319

Rigby, R. A. and Stasinopoulos, D. M. (2004). Smooth centile curves for skew and kurtotic data modelled using the Box-Cox Power
Exponential distribution. *Statistics in Medicine*, **23**: 3053-3076.

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.

##### See Also

##### Examples

```
# NOT RUN {
BCCG() # gives information about the default links for the Cole and Green distribution
# library(gamlss)
#data(abdom)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=BCCG, data=abdom)
#plot(h)
plot(function(x) dBCCG(x, mu=5,sigma=.5,nu=-1), 0.0, 20,
main = "The BCCG density mu=5,sigma=.5,nu=-1")
plot(function(x) pBCCG(x, mu=5,sigma=.5,nu=-1), 0.0, 20,
main = "The BCCG cdf mu=5, sigma=.5, nu=-1")
# }
```

*Documentation reproduced from package gamlss.dist, version 5.1-7, License: GPL-2 | GPL-3*