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`

.

```
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)
```

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

`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.

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 \(\sigma>0\).

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,10.1201/9780429298547. An older version can be found in https://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, 10.18637/jss.v023.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.
10.1201/b21973

(see also https://www.gamlss.com/).

# 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) # }