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This paper considers a new generalization of the negative binomial distribution arising from the Poisson-stopped sum of the Hurwitz–Lerch zeta distributions (Poisson-HLZD). The proposed generalization has much flexibility in shape (high probability at zero and long tail) as compared to other generalizations but retains computational tractability since, as a Poisson-stopped sum, it has a recurrence formula for its probabilities which facilitates its computation and applications. The flexibility of the Poisson-HLZD makes it a useful model for statistical analysis. The Poisson-HLZD is proved to have a mixed Poisson formulation and some of its probabilistic properties that are useful in determining the parameter space in estimation are derived. Its usefulness in data-fitting are compared with some well-known count frequency models. It is shown that the Poisson-HLZD fits very well for data with high zero counts or with very long tail.

The Poisson-stopped Hurwitz–Lerch zeta distribution