# GA

##### Gamma distribution for fitting a GAMLSS

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`

.

- Keywords
- distribution, regression

##### Usage

```
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
$$f(y|\mu,\sigma)=\frac{y^{(1/\sigma^2-1)}\exp[-y/(\sigma^2 \mu)]}{(\sigma^2 \mu)^{(1/\sigma^2)} \Gamma(1/\sigma^2)}$$
for \(y>0\), \(\mu>0\) and \(\sigma>0\).

##### 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`

.

##### 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.

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 http://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, 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.

##### See Also

##### Examples

```
# NOT RUN {
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)
# }
```

*Documentation reproduced from package gamlss.dist, version 5.1-7, License: GPL-2 | GPL-3*