R: The zero adjusted Inverse Gaussian distribution for fitting a...

ZAIG {gamlss.dist} | R Documentation

## 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`

### 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)*(2*pi*sigma^2*y^3)^.5 exp(-(y-mu)^2/(2*mu^2sigma^2*y))

otherwise

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

### Value

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

### Author(s)

Bob Rigby and Mikis Stasinopoulos

### 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.

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`, `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]

[Package gamlss.dist version 5.0-2 Index]