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 (very old) 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.
BCPE(mu.link = "identity", sigma.link = "log", nu.link = "identity",
tau.link = "log")
BCPEo(mu.link = "log", sigma.link = "log", nu.link = "identity",
tau.link = "log")
BCPEuntr(mu.link = "identity", sigma.link = "log", nu.link = "identity",
tau.link = "log")
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,...)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.
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 \(\tau>0\) see pp. 450-451 of Rigby et al. (2019).
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.
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, tools:::Rd_expr_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, tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1201/b21973")
(see also https://www.gamlss.com/).
gamlss.family, BCT
# 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")
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