The function GG defines the generalized gamma distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The parameterization used has the mean of the distribution equal to mu and the variance equal to \((\sigma^2)(\mu^2)\). The functions dGG, pGG, qGG and rGG define the density, distribution function, quantile function and random generation for the specific parameterization of the generalized gamma distribution defined by function GG.
GG(mu.link = "log", sigma.link = "log",
nu.link = "identity")
dGG(x, mu=1, sigma=0.5, nu=1,
log = FALSE)
pGG(q, mu=1, sigma=0.5, nu=1, lower.tail = TRUE,
log.p = FALSE)
qGG(p, mu=1, sigma=0.5, nu=1, lower.tail = TRUE,
log.p = FALSE )
rGG(n, mu=1, sigma=0.5, nu=1)GG() returns a gamlss.family object which can be used to fit a generalized gamma distribution in the gamlss() function. dGG() gives the density, pGG() gives the distribution function, qGG() gives the quantile function, and rGG() generates random deviates.
Defines the mu.link, with "log" link as the default for the mu parameter, other links are "inverse" and "identity"
Defines the sigma.link, with "log" link as the default for the sigma parameter, other links are "inverse" and "identity"
Defines the nu.link, with "identity" link as the default for the sigma parameter, other links are \(1/nu^2\) and "log"
vector of quantiles
vector of location parameter values
vector of scale parameter values
vector of shape 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
Mikis Stasinopoulos, Bob Rigby and Nicoleta Motpan
The specific parameterization of the generalized gamma distribution used in GG is $$f(y|\mu,\sigma,\nu)= \frac{\theta^\theta z^\theta |\nu| e^(-\theta z)}{(\Gamma(\theta)y)}$$ where \(z =(y/\mu)^\nu\), \(\theta = 1/(\sigma^2|\nu|^2)\) for y>0, \(\mu>0\), \(\sigma>0\) and \(-\infty<\nu<+\infty\) see pp. 443-444 of Rigby et al. (2019). Note that for \(\nu=0\) the distribution is log normal.
Lopatatzidis, A. and Green, P. J. (2000), Nonparametric quantile regression using the gamma distribution, unpublished.
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., 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, GA
y<-rGG(100,mu=1,sigma=0.1, nu=-.5) # generates 100 random observations
hist(y)
# library(gamlss)
#histDist(y, family=GG)
#m1 <-gamlss(y~1,family=GG)
#prof.dev(m1, "nu", min=-2, max=2, step=0.2)
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