gamlss.dist (version 6.1-1)

ZIP: Zero inflated poisson distribution for fitting a GAMLSS model

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

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)

Value

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

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]

Author

Mikis Stasinopoulos, Bob Rigby

Details

Let \(Y=0\) with probability \(\sigma\) and \(Y \sim Po(\mu)\) with probability \((1-\sigma)\) the Y has a Zero inflated Poisson Distribution given by

$$f(y)=\sigma +(1-\sigma)e^{-\mu}$$ if (y=0) $$f(y)=(1-\sigma)\frac{e^{-\mu} \mu^y}{y!}$$ if (y>0) for \(y=0,1,...\) see pp 498-500 of Rigby et al. (2019). .

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, tools:::Rd_expr_doi("10.1201/9780429298547"). 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, tools:::Rd_expr_doi("10.18637/jss.v023.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. tools:::Rd_expr_doi("10.1201/b21973")

(see also https://www.gamlss.com/).

See Also

gamlss.family, PO, ZIP2

Examples

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]