R: Inverse Gamma distribution for fitting a GAMLSS

IGAMMA {gamlss.dist} | R Documentation

## Inverse Gamma distribution for fitting a GAMLSS

### 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

    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.

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. 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, 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`

### Examples

    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"))

[Package gamlss.dist version 5.3-2 Index]