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.

Usage

BCT(mu.link = "identity", 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)

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

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 3.1-0 Index]