ZIP2: 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 `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``` ```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.

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`, `ZIP`
 ```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] ```