R: Box-Cox t distribution for fitting a GAMLSS

BCT {gamlss.dist} | R Documentation

## Box-Cox t distribution for fitting a GAMLSS

### Description

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.

### Usage

    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)

### 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", "own"
`nu.link` | Defines the `nu.link`, with "identity" link as the default for the `nu` parameter. Other links are "inverse", "log", "own"
`tau.link` | Defines the `tau.link`, with "log" link as the default for the `tau` parameter. Other links are "inverse", "identity" and "own"
`x,q` | vector of quantiles
`mu` | vector of location parameter values
`sigma` | vector of scale parameter values
`nu` | vector of `nu` parameter values
`tau` | vector of `tau` 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 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.

### Value

`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.

### Warning

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 ]

### Note

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

### Author(s)

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

### References

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`, `BCCG`

### Examples

    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")

[Package gamlss.dist version 5.0-2 Index]