GIG: 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, ...)

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.

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

Author

Mikis Stasinopoulos, Bob Rigby and Nicoleta Motpan

Details

The specific parameterization of the generalized inverse gaussian distribution used in GIG is $$f(y|\mu,\sigma,\nu)= (\frac{b}{\mu})^\nu \left[ \frac{y^{\nu-1}}{2 K_{\nu}(\sigma^{-2}) }\right] \exp \left[-\frac{1}{2\sigma^2} \left(\frac{b y }{\mu}+\frac{\mu}{ b y} \right)\right]$$ where $b = \frac{K_{\nu+1}(\frac{1}{\sigma^2})}{K_{\nu}(\frac{1}{\sigma^{-2}})}$, for y>0, $\mu>0$, $\sigma>0$ and $-\infty<\nu<+\infty$ see pp 445-446 of Rigby et al. (2019).

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, tools:::Rd_expr_doi("10.1201/9780429298547"). 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, tools:::Rd_expr_doi("10.18637/jss.v023.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. tools:::Rd_expr_doi("10.1201/b21973")

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

(see also https://www.gamlss.com/).

Examples

Run this code

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

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