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Characterizing Existence and Location of the ML Estimate in the Conway–Maxwell–Poisson Model

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Abstract

As a flexible extension of the common Poisson model, the Conway–Maxwell–Poisson distribution allows for describing under- and overdispersion in count data via an additional parameter. Estimation methods for two Conway–Maxwell–Poisson parameters are then required to specify the model. In this work, two characterization results are provided related to maximum likelihood estimation of the Conway–Maxwell–Poisson parameters. The first states that maximum likelihood estimation fails if and only if the range of the observations is less than two. Assuming that the maximum likelihood estimate exists, the second result then comprises a simple necessary and sufficient condition for the maximum likelihood estimate to be a solution of the likelihood equation; otherwise it lies on the boundary of the parameter set. A simulation study is carried out to investigate the accuracy of the maximum likelihood estimate in dependence of the range of the underlying observations.

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Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. Institute of Statistics, RWTH Aachen University, 52056, Aachen, Germany

    Stefan Bedbur, Anton Imm & Udo Kamps

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  1. Stefan Bedbur
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  2. Anton Imm
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  3. Udo Kamps
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Correspondence to Stefan Bedbur.

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Bedbur, S., Imm, A. & Kamps, U. Characterizing Existence and Location of the ML Estimate in the Conway–Maxwell–Poisson Model. Math. Meth. Stat. 33, 70–78 (2024). https://doi.org/10.3103/S1066530724700042

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  • DOI: https://doi.org/10.3103/S1066530724700042

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