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.
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,...)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"
vector of quantiles
vector of location parameter values
vector of scale parameter values
vector of nu parameter values
vector of tau parameter values
logical; if TRUE, probabilities p are given as log(p).
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
vector of probabilities.
number of observations. If length(n) > 1, the length is
taken to be the number required
a gamlss BCPE family object
for extra arguments
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 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. \(\mu>0,\sigma>0, \nu=(-\infty,\infty) {\rm and} \tau>0\) ]
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.
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.
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.
# 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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