BCCG | R Documentation |

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.

```
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)
```

`mu.link` |
Defines the |

`sigma.link` |
Defines the |

`nu.link` |
Defines the |

`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 |

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 and 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) or pp. 439-441 of Rigby et al. (2019) for details.

`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.

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)`

]

`\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)`

]

Mikis Stasinopoulos, Bob Rigby and Kalliope Akantziliotou

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. *Statistical Modelling*, 6(3) :209. \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`

, `BCT`

```
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")
```

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