The zeroinflated negative binomial (ZINB) model in PROC TCOUNTREG 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.
In this model, the probability is given by the logistic function—namely,

The loglikelihood function is












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














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





















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





