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

## Description

The function `GA` defines the gamma distribution, a two 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 `dGA`, `pGA`, `qGA` and `rGA` define the density, distribution function, quantile function and random generation for the specific parameterization of the gamma distribution defined by function `GA`.

## Usage

 ```1 2 3 4 5``` ```GA(mu.link = "log", sigma.link ="log") dGA(x, mu = 1, sigma = 1, log = FALSE) pGA(q, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) qGA(p, mu = 1, sigma = 1, lower.tail = TRUE, log.p = FALSE) rGA(n, mu = 1, sigma = 1) ```

## Arguments

 `mu.link` Defines the `mu.link`, with "log" link as the default for the mu parameter, other links are "inverse", "identity" ans "own" `sigma.link` Defines the `sigma.link`, with "log" link as the default for the sigma parameter, other link is the "inverse", "identity" and "own" `x,q` vector of quantiles `mu` vector of location parameter values `sigma` vector of scale 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 gamma distribution used in `GA` is

f(y|mu,sigma)=(y^((1/sigma^2)-1)*exp[-y/((sigma^2)*mu)])/((sigma^2*mu)^(1/sigma^2) Gamma(1/sigma^2))

for y>0, μ>0 and σ>0.

## Value

`GA()` returns a `gamlss.family` object which can be used to fit a gamma distribution in the `gamlss()` function. `dGA()` gives the density, `pGA()` gives the distribution function, `qGA()` gives the quantile function, and `rGA()` generates random deviates. The latest functions are based on the equivalent `R` functions for gamma distribution.

## Note

mu is the mean of the distribution in `GA`. In the function `GA`, sigma is the square root of the usual dispersion parameter for a GLM gamma model. Hence sigma*mu is the standard deviation of the distribution defined in `GA`.

## Author(s)

Mikis Stasinopoulos [email protected], Bob Rigby and Calliope Akantziliotou

## 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.

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`
 ```1 2 3 4 5 6 7 8``` ```GA()# gives information about the default links for the gamma distribution # dat<-rgamma(100, shape=1, scale=10) # generates 100 random observations # fit a gamlss model # gamlss(dat~1,family=GA) # fits a constant for each parameter mu and sigma of the gamma distribution newdata<-rGA(1000,mu=1,sigma=1) # generates 1000 random observations hist(newdata) rm(dat,newdata) ```