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