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August 21, 2025

New mixture distributions for modelling count data

Key Points

The introduction of new mixture distributions provides enhanced modeling options for count data, showing increased accuracy in fitting datasets.
These distributions generate generalized forms of the Poisson distribution that can handle both under and over-dispersion effectively.
Probabilities from these models are straightforward to compute, facilitating random number generation and moments calculation.
Inference with these new distributions significantly sharpens compared to traditional Poisson models, highlighting improved results against standard benchmarks.

Abstract

A class of new 1-parameter underdispersed distributions is introduced. Mixed with Poisson distributions; they generate 2- and 3-parameter discrete distributions that generalize the Poisson distribution and can be both under and over-dispersed. Probabilities are easy to compute and moments and random number generation are tractable. The distributions are described, and they are fitted to some underdispersed and overdispersed datasets. We show how inference for the effect of covariates sharpens on moving from the Poisson model. The fits compare favourably to two benchmarks, the COM Poisson distribution and the weighted Poisson distribution.

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