R: Box-Cox Cole and Green distribution (or Box-Cox normal) for...

BCCG {gamlss.dist} | R Documentation

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

### Description

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.

### 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)=(1/(sqrt(2*pi)*sigma))*(y^(nu-1)/mu^nu)*exp(-z^2/2)

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

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.

### 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=(-Inf,+Inf)]

### 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, and nu=(-Inf,+Inf) ]

### Author(s)

Mikis Stasinopoulos mikis.stasinopoulos@gamlss.org, Bob Rigby and Kalliope Akantziliotou

### 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.

Stasinopoulos D. M., Rigby R.A. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://www.gamlss.org/).

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

`gamlss.family`, `BCPE`, `BCT`

### Examples

    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")

[Package gamlss.dist version 5.0-2 Index]