# BCt: Box-Cox t distribution for fitting a GAMLSS In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

 BCT 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 `BCTo` is identical to `BCT` but with log link for mu.

### Usage

```BCT(mu.link = "identity", sigma.link = "log", nu.link = "identity",
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 see pp. 450-451 of Rigby et al. (2019).

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. Statistical Modelling 6(3):200. 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, 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, 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. doi: 10.1201/b21973

`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")
```

gamlss.dist documentation built on Aug. 28, 2022, 5:05 p.m.