ZIP2: 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)

vector of (non-negative integer) quantiles

vector of positive means

sigma

vector of probabilities at zero

vector of probabilities

vector of quantiles

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]

Value

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

Details

Let $Y=0$ with probability $\sigma$ and $Y \sim Po(\mu/\left[1-\sigma \right])$ with probability $(1-\sigma)$ then Y has a Zero inflated Poisson type 2 distribution given by

$$f(y|\mu,\sigma)=\sigma +(1-\sigma)e^{-\mu/(1-\sigma)} \hspace{2mm} \mbox{if $y=0$} $$ $$f(y|\mu,\sigma)=(1-\sigma)\frac{e^{-\mu/(1-\sigma)} \left[\mu/(1-\sigma)\right]^y}{y!} \hspace{2mm} \mbox{if $y=1,2,3,\ldots$}$$

The mean of the distribution in this parameterization is mu.

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 http://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, 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.

Examples

Run this code

# NOT RUN {
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]
# }

Run the code above in your browser using DataCamp Workspace