Description Usage Arguments Details Value Note Author(s) References See Also Examples

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.

1 2 3 4 5 6 7 |

`mu.link` |
Defines the |

`sigma.link` |
Defines the |

`nu.link` |
Defines the |

`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 |

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

is:

*f(y|μ,σ_1)=(y^((1/σ_1^2)-1)*exp[-y/((σ_1^2)*μ)])/((σ_1^2*μ)^(1/σ_1^2) Gamma(1/σ_1^2)) *

for *y>0*, *μ>0* where *σ_1=σ μ^(ν/1-1) *.

`GAF()`

returns a `gamlss.family`

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

function.

For the function `GAF()`

, *μ* is the mean and *σ μ^{ν/2}*
is the standard deviation of the gamma family.
The `GAF`

is design for fitting regression type models where the variance is proportional to a power of the mean.

Note that because the high correlation between the `sigma`

and the `nu`

parameter the `mixed()`

method should be used in the fitting.

Mikis Stasinopoulos, Robert Rigby and Fernanda De Bastiani

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.

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 | ```
GAF()
## Not run:
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)
## End(Not run)
``` |

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