GA | R Documentation |

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 |

`sigma.link` |
Defines the |

`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 |

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`

, see pp. 423-424 of Rigby et al. (2019).

`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.

`\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`

.

Mikis Stasinopoulos, Bob Rigby and Calliope Akantziliotou

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,\Sexpr[results=rd]{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, \Sexpr[results=rd]{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.
\Sexpr[results=rd]{tools:::Rd_expr_doi("10.1201/b21973")}

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

`gamlss.family`

```
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)
```

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