On estimation of the Poisson parameter in zero-modified Poisson models

For count data, typically, a Poisson model is assumed. However, it has been observed in various applications that this model does not fit, because of too few or too many zeros in the data. For the latter case the zero-inflated Poisson (regression) model has been proposed. In situations where, for certain reasons, no zeros at all can be observed, the zero-truncated Poisson (regression) model is appropriate. This paper provides an EM algorithm for ML estimation of the latter model by standard software for Poisson regression. It is shown, that this algorithm can be used also to estimate the Poisson parameters of zero-inflated, zero-deflated, and standard Poisson models, when the zero observations are ignored. If nothing is known about the kind of zero modification, the respective estimates are fully efficient. Situations are described, in which the loss of efficiency is small although knowledge of the kind of zero modification is available. In a second step, the respective estimates of the Poisson parameter can be used to analyze the kind of zero modification and to estimate the zero modification parameter.