Characterization of count data distributions involving additivity and binomial subsampling

An important problem in data analysis is how to choose an adequate family of distributions or statistical model to describe the values observed in any study. For this purpose the characterization theorems can be useful because, under general reasonable suppositions related to the nature of the experiment or the data, they allow us to reduce the possible set of distributions that can be used. One of these reasonable assumptions is that the model is closed under addition, that is, closed under convolutions. This means that the distribution of the sum of independent random variables with distributions belonging to the model also belongs to the same model. The property of closure under addition has widely been used to characterize families of distributions, the papers of Teicher [16] and Godambe and Patil [4] being particularly noteworthy. Other properties that can be utilized in characterization theorems jointly with additivity involve assumptions about the maximum likelihood estimators of the parameters. Puig [13] and Puig and Valero [14] characterize count families of distributions using several notions of additivity and assuming that the maximum likelihood estimator of the population mean is the sample mean.