# IGAMMA: Inverse Gamma distribution for fitting a GAMLSS In mstasinopoulos/GAMLSS-Distibutions: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

The function `IGAMMA()` defines the Inverse Gamma distribution, a two parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`, with parameters `mu` (the mode) and `sigma`. The functions `dIGAMMA`, `pIGAMMA`, `qIGAMMA` and `rIGAMMA` define the density, distribution function, quantile function and random generation for the `IGAMMA` parameterization of the Inverse Gamma distribution.

## Usage

 ```1 2 3 4 5``` ```IGAMMA(mu.link = "log", sigma.link="log") dIGAMMA(x, mu = 1, sigma = .5, log = FALSE) pIGAMMA(q, mu = 1, sigma = .5, lower.tail = TRUE, log.p = FALSE) qIGAMMA(p, mu = 1, sigma = .5, lower.tail = TRUE, log.p = FALSE) rIGAMMA(n, mu = 1, sigma = .5) ```

## Arguments

 `mu.link` Defines the `mu.link`, with `log` link as the default for the mu parameter `sigma.link` Defines the `sigma.link`, with `log` as the default for the sigma parameter `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 parameterization of the Inverse Gamma distribution in the function `IGAMMA` is

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

where alpha = 1/(sigma^2) for y>0, mu>0 and sigma>0.

## Value

returns a gamlss.family object which can be used to fit an Inverse Gamma distribution in the `gamlss()` function.

## Note

For the function `IGAMMA()`, mu is the mode of the Inverse Gamma distribution.

## Author(s)

Fiona McElduff, Bob Rigby and Mikis Stasinopoulos.

## 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.

`gamlss.family`, `GA`
 ``` 1 2 3 4 5 6 7 8 9 10``` ```par(mfrow=c(2,2)) y<-seq(0.2,20,0.2) plot(y, dIGAMMA(y), type="l") q <- seq(0.2, 20, 0.2) plot(q, pIGAMMA(q), type="l") p<-seq(0.0001,0.999,0.05) plot(p , qIGAMMA(p), type="l") dat <- rIGAMMA(50) hist(dat) #summary(gamlss(dat~1, family="IGAMMA")) ```