Gamma distribution for fitting a GAMLSS

The function GA defines the gamma distribution, a two parameter distribution, for a 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.

distribution, regression
GA( = "log", ="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)

Defines the, with "log" link as the default for the mu parameter, other links are "inverse", "identity" ans "own"

Defines the, with "log" link as the default for the sigma parameter, other link is the "inverse", "identity" and "own"


vector of quantiles


vector of location parameter values


vector of scale parameter values

log, log.p

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


The specific parameterization of the gamma distribution used in GA is $$f(y|\mu,\sigma)=\frac{y^{(1/\sigma^2-1)}\exp[-y/(\sigma^2 \mu)]}{(\sigma^2 \mu)^{(1/\sigma^2)} \Gamma(1/\sigma^2)}$$ for \(y>0\), \(\mu>0\) and \(\sigma>0\).


GA() returns a 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.


\(\mu\) 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.


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. An older version can be found in

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,

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.

See Also

  • GA
  • dGA
  • qGA
  • pGA
  • rGA
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
# }
Documentation reproduced from package gamlss.dist, version 5.1-7, License: GPL-2 | GPL-3

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