gamlss.dist (version 5.0-0)

ZAIG: The zero adjusted Inverse Gaussian distribution for fitting a GAMLSS model

Description

The function ZAIG() defines the zero adjusted Inverse Gaussian distribution, a three parameter distribution, for a gamlss.family object to be used in GAMLSS fitting using the function gamlss(). The zero adjusted Inverse Gaussian distribution is similar to the Inverse Gaussian distribution but allows zeros as y values. The extra parameter models the probabilities at zero. The functions dZAIG, pZAIG, qZAIG and rZAIG define the density, distribution function, quantile function and random generation for the ZAIG parameterization of the zero adjusted Inverse Gaussian distribution. plotZAIG can be used to plot the distribution. meanZAIG calculates the expected value of the response for a fitted model.

Usage

ZAIG(mu.link = "log", sigma.link = "log", nu.link = "logit")
dZAIG(x, mu = 1, sigma = 1, nu = 0.1, log = FALSE)
pZAIG(q, mu = 1, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE)
qZAIG(p, mu = 1, sigma = 1, nu = 0.1, lower.tail = TRUE, log.p = FALSE, 
     upper.limit = mu + 10 * sqrt(sigma^2 * mu^3))
rZAIG(n, mu = 1, sigma = 1, nu = 0.1, ...)
plotZAIG(mu = 5, sigma = 1, nu = 0.1, from = 0, to = 10, n = 101, ...)
meanZAIG(obj)

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" link as the default for the sigma parameter

nu.link

Defines the nu.link, with "logit" link as the default for the sigma parameter

x,q

vector of quantiles

mu

vector of location parameter values

sigma

vector of scale parameter values

nu

vector of probability at zero 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]

upper.limit

the argument upper.limit sets the upper limit in the golden section search for q, the default is 10 time its standard deviation

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required

from

where to start plotting the distribution from

to

up to where to plot the distribution

obj

a fitted BEINF object

…

… can be used to pass the uppr.limit argument to qIG

Value

returns a gamlss.family object which can be used to fit a zero adjusted inverse Gaussian distribution in the gamlss() function.

Details

The Zero adjusted IG distribution is given as $$f(y|\mu,\sigma\,\nu)=\nu$$ if (y=0) $$f(y|\mu,\sigma,\nu)=(1-\nu)\frac{1}{\sqrt{2 \pi \sigma^2 y^3}} \exp(- \frac{(y-\mu)^2}{2\mu^2\sigma^2y})$$ otherwise

for \(y=(0,\infty)\), \(\mu>0\), \(\sigma>0\) and \(0< \nu< 1\). \(E(y)=(1-\nu)\mu\) and \(Var(y)=(1-\nu)\mu^2(\nu+\mu\sigma^2)\).

References

Heller, G. Stasinopoulos M and Rigby R.A. (2006) The zero-adjusted Inverse Gaussian distribution as a model for insurance claims. in Proceedings of the 21th International Workshop on Statistial Modelling, eds J. Hinde, J. Einbeck and J. Newell, pp 226-233, Galway, Ireland.

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.

See Also

gamlss.family, IG

Examples

ZAIG()# gives information about the default links for the ZAIG distribution
# plotting the distribution
plotZAIG( mu =10 , sigma=.5, nu = 0.1, from = 0, to=10, n = 101)
# plotting the cdf
plot(function(y) pZAIG(y, mu=10 ,sigma=.5, nu = 0.1 ), 0, 1)
# plotting the inverse cdf
plot(function(y) qZAIG(y, mu=10 ,sigma=.5, nu = 0.1 ), 0.001, .99)
# generate random numbers
dat <- rZAIG(100,mu=10,sigma=.5, nu=.1)
# fit a model to the data 
# library(gamlss)
# m1<-gamlss(dat~1,family=ZAIG)
# meanZAIG(m1)[1]