# ZIP: Zero inflated poisson distribution for fitting a GAMLSS model In gamlss.dist: Distributions for Generalized Additive Models for Location Scale and Shape

## Description

The function `ZIP` defines the zero inflated Poisson distribution, a two parameter distribution, for a `gamlss.family` object to be used in GAMLSS fitting using the function `gamlss()`. The functions `dZIP`, `pZIP`, `qZIP` and `rZIP` define the density, distribution function, quantile function and random generation for the inflated poisson, `ZIP()`, distribution.

## Usage

 ```1 2 3 4 5``` ```ZIP(mu.link = "log", sigma.link = "logit") dZIP(x, mu = 5, sigma = 0.1, log = FALSE) pZIP(q, mu = 5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE) qZIP(p, mu = 5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE) rZIP(n, mu = 5, sigma = 0.1) ```

## Arguments

 `mu.link` defines the `mu.link`, with "log" link as the default for the `mu` parameter `sigma.link` defines the `sigma.link`, with "logit" link as the default for the sigma parameter which in this case is the probability at zero. Other links are "probit" and "cloglog"'(complementary log-log) `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 Y \sim Po(μ) with probability (1-σ) the Y has a Zero inflated Poisson Distribution given by

sigma+(1-sigma)e^(-mu)

if (y=0)

f(y)=(1-sigma)e^-mu mu^y/y!

if (y>0) for y=0,1,...,.

## Value

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

## Author(s)

Mikis Stasinopoulos [email protected], Bob Rigby

## 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.

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.

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