# Zero inflated poisson distribution for fitting a GAMLSS model

### Description

The function `ZIP2`

defines the zero inflated Poisson type 2 distribution, a two parameter distribution, for a `gamlss.family`

object to be used in GAMLSS fitting
using the function `gamlss()`

. The functions `dZIP2`

, `pZIP2`

, `qZIP2`

and `rZIP2`

define the density, distribution function, quantile function
and random generation for the inflated poisson, `ZIP2()`

, distribution.
The ZIP2 is a different parameterization of the ZIP distribution. In the ZIP2 the `mu`

is the mean of the distribution.

### Usage

1 2 3 4 5 |

### Arguments

`mu.link` |
defines the |

`sigma.link` |
defines the |

`x` |
vector of (non-negative integer) quantiles |

`mu` |
vector of positive means |

`sigma` |
vector of probabilities at zero |

`p` |
vector of probabilities |

`q` |
vector of quantiles |

`n` |
number of random values to return |

`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] |

### Details

Let *Y=0* with probability *σ* and *Po(mu/(1-sigma))*
with probability *(1-σ)* then Y has
a Zero inflated Poisson type 2 distribution given by

*sigma+(1-sigma)e^(-(mu/(1-sigma))) if y=0*

*f(y)=(1-sigma)exp(-(mu/(1-sigma)))* (mu/(1-sigma))^y/y! if y=0,1,2,...*

The mean of the distribution in this parameterization is `mu`

.

### Value

returns a `gamlss.family`

object which can be used to fit a zero inflated poisson distribution in the `gamlss()`

function.

### Author(s)

Bob Rigby, Gillian Heller and Mikis Stasinopoulos

### References

Lambert, D. (1992), Zero-inflated Poisson Regression with an application to defects in Manufacturing,
*Technometrics*, **34**, pp 1-14.

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`

, `ZIP`

### Examples

1 2 3 4 5 6 7 8 9 | ```
ZIP2()# gives information about the default links for the normal distribution
# creating data and plotting them
dat<-rZIP2(1000, mu=5, sigma=.1)
r <- barplot(table(dat), col='lightblue')
# fit the disteibution
# library(gamlss)
# mod1<-gamlss(dat~1, family=ZIP2)# fits a constant for mu and sigma
# fitted(mod1)[1]
# fitted(mod1,"sigma")[1]
``` |