The function `GAF()`

defines a gamma distribution family, which has three parameters. This is not the generalised gamma distribution which is called `GG`

. The third parameter here is to model the mean and variance relationship. The distribution can be fitted using the function `gamlss()`

. The mean of `GAF`

is equal to `mu`

. The variance is equal to `sigma^2*mu^nu`

so the standard deviation is `sigma*mu^(nu/2)`

. The function is design for cases where the variance is proportional to a power of the mean. This is an instance of the Taylor's power low, see Enki et al. (2017). The functions `dGAF`

, `pGAF`

, `qGAF`

and `rGAF`

define the density, distribution function,
quantile function and random generation for the `GAF`

parametrization of the gamma family.

```
GAF(mu.link = "log", sigma.link = "log", nu.link = "identity")
dGAF(x, mu = 1, sigma = 1, nu = 2, log = FALSE)
pGAF(q, mu = 1, sigma = 1, nu = 2, lower.tail = TRUE,
log.p = FALSE)
qGAF(p, mu = 1, sigma = 1, nu = 2, lower.tail = TRUE,
log.p = FALSE)
rGAF(n, mu = 1, sigma = 1, nu = 2)
```

`GAF()`

returns a `gamlss.family`

object which can be used to fit the gamma family in the `gamlss()`

function.

- mu.link
Defines the

`mu.link`

, with "identity" link as the default for the mu parameter- sigma.link
Defines the

`sigma.link`

, with "log" link as the default for the sigma parameter- nu.link
Defines the

`nu.link`

with "identity" link as the default for the nu parameter- x,q
vector of quantiles

- mu
vector of location parameter values

- sigma
vector of scale parameter values

- nu
vector of power 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

Mikis Stasinopoulos, Robert Rigby and Fernanda De Bastiani

The parametrization of the gamma family given in the function `GAF()`

is:

$$f(y|\mu,\sigma_1)=\frac{y^{(1/\sigma_1^2-1)}\exp[-y/(\sigma_1^2 \mu)]}{(\sigma_1^2 \mu)^{(1/\sigma^2)} \Gamma(1/\sigma^2)}$$ for \(y>0\), \(\mu>0\) where \(\sigma_1=\sigma \mu^{(\nu/2)-1} \) \(\sigma>0\) and \(-\infty <\nu< \infty\) see pp. 442-443 of Rigby et al. (2019).

Enki, D G, Noufaily, A., Farrington, P., Garthwaite, P., Andrews, N. and Charlett, A. (2017) Taylor's power law and the statistical modelling of infectious disease surveillance data, Journal of the Royal Statistical Society: Series A (Statistics in Society), volume=180, number=1, pages=45-72.

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, tools:::Rd_expr_doi("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, tools:::Rd_expr_doi("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. tools:::Rd_expr_doi("10.1201/b21973")

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

`gamlss.family`

, `GA`

, `GG`

```
GAF()
if (FALSE) {
m1<-gamlss(y~poly(x,2),data=abdom,family=GAF, method=mixed(1,100),
c.crit=0.00001)
# using RS()
m2<-gamlss(y~poly(x,2),data=abdom,family=GAF, n.cyc=5000, c.crit=0.00001)
# the estimates of nu slightly different
fitted(m1, "nu")[1]
fitted(m2, "nu")[1]
# global deviance almost identical
AIC(m1, m2)
}
```

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