R: Generalized Inverse Gaussian distribution for fitting a...

GIG {gamlss.dist} | R Documentation

## Generalized Inverse Gaussian distribution for fitting a GAMLSS

### Description

The function GIG defines the generalized inverse gaussian distribution, a three parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `DIG`, `pGIG`, `GIG` and `rGIG` define the density, distribution function, quantile function and random generation for the specific parameterization of the generalized inverse gaussian distribution defined by function GIG.

### Usage

    GIG(mu.link = "log", sigma.link = "log",
                           nu.link = "identity")
    dGIG(x, mu=1, sigma=1, nu=1,
                          log = FALSE)
    pGIG(q, mu=1, sigma=1, nu=1,  lower.tail = TRUE,
                         log.p = FALSE)
    qGIG(p, mu=1, sigma=1, nu=1,  lower.tail = TRUE,
                         log.p = FALSE)
    rGIG(n, mu=1, sigma=1, nu=1, ...)

### Arguments

`mu.link` | Defines the mu.link, with "log" link as the default for the `mu` parameter, other links are "inverse" and "identity"
`sigma.link` | Defines the sigma.link, with "log" link as the default for the `sigma` parameter, other links are "inverse" and "identity"
`nu.link` | Defines the nu.link, with "identity" link as the default for the `nu` parameter, other links are "inverse" and "log"
`x,q` | vector of quantiles
`mu` | vector of location parameter values
`sigma` | vector of scale parameter values
`nu` | vector of shape 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
`...` | for extra arguments

### Details

The specific parameterization of the generalized inverse gaussian distribution used in GIG is f(y|mu,sigma,nu)=((c/mu)^nu)*(y^(nu-1))/(2*besselK(1/sigma,nu))*exp(-1/(2*sigma)*(c*y/mu+mu/(c*y))) where c = besselK(1/sigma,nu+1)/besselK(1/sigma,nu), for y>0, mu>0, sigma>0 and -Inf>nu>Inf.

### Value

GIG() returns a gamlss.family object which can be used to fit a generalized inverse gaussian distribution in the gamlss() function. DIG() gives the density, pGIG() gives the distribution function, GIG() gives the quantile function, and rGIG() generates random deviates.

### Author(s)

Mikis Stasinopoulos mikis.stasinopoulos@gamlss.org, Bob Rigby and Nicoleta Motpan

### 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.

Jorgensen B. (1982) Statistical properties of the generalized inverse Gaussian distribution, Series: Lecture notes in statistics; 9, New York : Springer-Verlag.

### See Also

`gamlss.family`, `IG`

### Examples

    y<-rGIG(100,mu=1,sigma=1, nu=-0.5) # generates 1000 random observations
    hist(y)
    # library(gamlss)
    # histDist(y, family=GIG)

[Package gamlss.dist version 5.0-2 Index]