# 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.