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On the Nielsen distribution

We introduce a two-parameter discrete distribution that may have a zero vertex and can be useful for modeling overdispersion. The discrete Nielsen distribution generalizes the Fisher logarithmic (i.e. logarithmic series) and Stirling type I distributions in the sense that both can be considered displacements of the Nielsen distribution. We provide a comprehensive account of the structural properties of the new discrete distribution. We also show that the Nielsen distribution is infinitely divisible. We discuss maximum likelihood estimation of the model parameters and provide a simple method to find them numerically. The usefulness of the proposed distribution is illustrated by means of three real data sets to prove its versatility in practical applications.