R: The Gamma distribution family

GAF {gamlss.dist} | R Documentation

## The Gamma distribution family

### Description

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.

### Usage

    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)

### Arguments

`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

### Details

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) .

### Value

`GAF()` returns a `gamlss.family` object which can be used to fit the gamma family in the `gamlss()` function.

### Note

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.

### Author(s)

Mikis Stasinopoulos, Robert Rigby and Fernanda De Bastiani

### References

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, https://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

`gamlss.family`, `GA`, `GG`

### Examples

    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)

[Package gamlss.dist version 5.3-2 Index]