R: Zero inflated poisson distribution for fitting a GAMLSS model

ZIP2 {gamlss.dist} | R Documentation

## 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

    ZIP2(mu.link = "log", sigma.link = "logit")
    dZIP2(x, mu = 5, sigma = 0.1, log = FALSE)
    pZIP2(q, mu = 5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE)
    qZIP2(p, mu = 5, sigma = 0.1, lower.tail = TRUE, log.p = FALSE)
    rZIP2(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 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.

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. 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, https://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`, `ZIP`

### Examples

    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]

[Package gamlss.dist version 5.3-2 Index]