In the Conway-Maxwell-Poisson regression model, the data generation process is defined as
where the normalization factor is
and
The zero-inflated Conway-Maxwell-Poisson model can be written as
The conditional expectation and conditional variance of are given by
General form of the log-likelihood function for the Conway-Maxwell-Poisson zero-inflated model is
In this model the probability is expressed with a logistic link function as
The log-likelihood function is