In the zeroinflated Poisson (ZIP) regression model, the data generation process 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 since .
In general, the loglikelihood 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 loglikelihood function and the gradient.
First, consider the ZIP model in which the probability is expressed with a logistic link function—namely,

The loglikelihood function is









See 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 with a standard normal link function: . The loglikelihood function is






See Poisson Regression for the definition of .
The gradient for this model is given by











