R: Gamma distribution for fitting a GAMLSS

GA {gamlss.dist} | R Documentation

## Gamma distribution for fitting a GAMLSS

### Description

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`.

### Usage

    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)

### Arguments

`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

### Details

The specific parameterization of the gamma distribution used in `GA` is

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

for y>0, μ>0 and σ>0.

### Value

`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.

### Note

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`.

### Author(s)

Mikis Stasinopoulos mikis.stasinopoulos@gamlss.org, Bob Rigby and Calliope Akantziliotou

### References

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. and Akantziliotou C. (2006) Instructions on how to use the GAMLSS package in R. Accompanying documentation in the current GAMLSS help files, (see also http://www.gamlss.org/).

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.

### See Also

`gamlss.family`

### Examples

    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)

[Package gamlss.dist version 5.0-2 Index]