gamlss.dist (version 6.0-1)

# BCPE: Box-Cox Power Exponential distribution for fitting a GAMLSS

## Description

This function defines the Box-Cox Power Exponential distribution, a four parameter distribution, for a gamlss.family object to be used for a GAMLSS fitting using the function gamlss(). The functions dBCPE, pBCPE, qBCPE and rBCPE define the density, distribution function, quantile function and random generation for the Box-Cox Power Exponential distribution. The function checkBCPE can be used, typically when a BCPE model is fitted, to check whether there exit a turning point of the distribution close to zero. It give the number of values of the response below their minimum turning point and also the maximum probability of the lower tail below minimum turning point. [The function Biventer() is the original version of the function suitable only for the untruncated BCPE distribution.] See Rigby and Stasinopoulos (2003) for details. The function BCPEo is identical to BCPE but with log link for mu.

## Usage

BCPE(mu.link = "identity", sigma.link = "log", nu.link = "identity",
dBCPE(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCPE(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
qBCPE(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE, log.p = FALSE)
rBCPE(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)
dBCPEo(x, mu = 5, sigma = 0.1, nu = 1, tau = 2, log = FALSE)
pBCPEo(q, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE,
log.p = FALSE)
qBCPEo(p, mu = 5, sigma = 0.1, nu = 1, tau = 2, lower.tail = TRUE,
log.p = FALSE)
rBCPEo(n, mu = 5, sigma = 0.1, nu = 1, tau = 2)
checkBCPE(obj = NULL, mu = 10, sigma = 0.1, nu = 0.5, tau = 2,...)

## Arguments

Defines the mu.link, with "identity" link as the default for the mu parameter. Other links are "inverse", "log" and "own"

Defines the sigma.link, with "log" link as the default for the sigma parameter. Other links are "inverse", "identity" and "own"

Defines the nu.link, with "identity" link as the default for the nu parameter. Other links are "inverse", "log" and "own"

Defines the tau.link, with "log" link as the default for the tau parameter. Other links are "logshifted", "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

obj

a gamlss BCPE family object

for extra arguments

## Value

BCPE() returns a gamlss.family object which can be used to fit a Box Cox Power Exponential distribution in the gamlss() function. dBCPE() gives the density, pBCPE() gives the distribution function, qBCPE() gives the quantile function, and rBCPE() generates random deviates.

## Warning

The BCPE.untr 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 BCPE distribution is suitable for all combinations of the parameters within their ranges [i.e. $$\mu>0,\sigma>0, \nu=(-\infty,\infty) {\rm and} \tau>0$$ ]

## Details

The probability density function of the untrucated Box Cox Power Exponential distribution, (BCPE.untr), is defined as

$$f(y|\mu,\sigma,\nu,\tau)=\frac{y^{\nu-1} \tau \exp[-\frac{1}{2}|\frac{z}{c}|^\tau]}{\mu^{\nu} \sigma c 2^{(1+1/\tau)} \Gamma(\frac{1}{\tau})}$$

where $$c = [ 2^{(-2/\tau)}\Gamma(1/\tau)/\Gamma(3/\tau)]^{0.5}$$, where if $$\nu \neq 0$$ then $$z=[(y/\mu)^{\nu}-1]/(\nu \sigma)$$ else $$z=\log(y/\mu)/\sigma$$, for $$y>0$$, $$\mu>0$$, $$\sigma>0$$, $$\nu=(-\infty,+\infty)$$ and $$\tau>0$$.

The Box-Cox Power Exponential, BCPE, 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.

## 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. 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., 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, 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, 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. 10.1201/b21973

gamlss.family, BCT

## Examples

# NOT RUN {
# BCPE()   #
# library(gamlss)
# data(abdom)
#h<-gamlss(y~cs(x,df=3), sigma.formula=~cs(x,1), family=BCPE, data=abdom)
#plot(h)
plot(function(x)dBCPE(x, mu=5,sigma=.5,nu=1, tau=3), 0.0, 15,
main = "The BCPE  density mu=5,sigma=.5,nu=1, tau=3")
plot(function(x) pBCPE(x, mu=5,sigma=.5,nu=1, tau=3), 0.0, 15,
main = "The BCPE  cdf mu=5, sigma=.5, nu=1, tau=3")
# }