The zero-inflated negative binomial (ZINB) model in PROC COUNTREG is based on the negative binomial model with quadratic variance function (p=2). The ZINB model is obtained by specifying a negative binomial distribution for the data generation process referred to earlier as Process 2:

Thus the ZINB model is defined to be

In this case, the conditional expectation and conditional variance of are

As with the ZIP model, the ZINB model exhibits overdispersion because the conditional variance exceeds the conditional mean.

ZINB Model with Logistic Link Function

In this model, the probability is given by the logistic function, namely

The log-likelihood function is

The gradient for this model is given by

ZINB Model with Standard Normal Link Function

For this model, the probability is specified with the standard normal distribution function (probit function): . The log-likelihood function is

The gradient for this model is given by