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.
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)
returns a gamlss.family
object which can be used to fit a zero inflated poisson distribution in the gamlss()
function.
defines the mu.link
, with "log" link as the default for the mu
parameter
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
vector of probabilities at zero
vector of probabilities
vector of quantiles
number of random values to return
logical; if TRUE, probabilities p are given as log(p)
logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x]
Bob Rigby, Gillian Heller and Mikis Stasinopoulos
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
.
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/).
gamlss.family
, ZIP
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 DataLab