In the zero-inflated Poisson (ZIP) regression model, the data generation process that is referred to earlier as Process 2 is
where . Thus the ZIP model is defined as
The conditional expectation and conditional variance of are given by
Note that the ZIP model (as well as the ZINB model) exhibits overdispersion because .
In general, the log-likelihood function of the ZIP model is
After a specific link function (either logistic or standard normal) for the probability is chosen, it is possible to write the exact expressions for the log-likelihood function and the gradient.
First, consider the ZIP model in which the probability is expressed using a logistic link function—namely,
The log-likelihood function is
See the section Poisson Regression for the definition of .
The gradient for this model is given by
Next, consider the ZIP model in which the probability is expressed using a standard normal link function: . The log-likelihood function is
See the section Poisson Regression for the definition of .
The gradient for this model is given by