# GG: Generalized Gamma distribution for fitting a GAMLSS In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

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.

## Usage

 ```1 2 3 4 5 6 7 8 9``` ```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) ```

## Arguments

 `mu.link` Defines the mu.link, with "log" link as the default for the mu parameter, other links are "inverse" and "identity" `sigma.link` Defines the sigma.link, with "log" link as the default for the sigma parameter, other links are "inverse" and "identity" `nu.link` Defines the nu.link, with "identity" link as the default for the sigma parameter, other links are "1/nu^2" and "log" `x,q` vector of quantiles `mu` vector of location parameter values `sigma` vector of scale parameter values `nu` vector of shape 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 specific parameterization of the generalized gamma distribution used in GG is

f(y|mu,sigma,nu)=theta^theta*z^theta*nu*e^(-theta*z)/(Gamma(theta)*y)

where z =(y/mu)^nu, theta = 1/(sigma^2*abs(nu)^2) for y>0, mu>0, sigma>0 and -Inf>nu>Inf. Note that for nu=0 the distribution is log normal.

## Value

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.

## Author(s)

Mikis Stasinopoulos, Bob Rigby and Nicoleta Motpan

## References

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.

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.

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.

`gamlss.family`, `GA`
 ```1 2 3 4 5 6``` ```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) ```