A Generalized Poisson–Gamma Hybrid Model for 
Heavy-Tailed Count Data

Isiak Kamaldeen Olomoda; Musa Yunus Olatunji

doi:doi:10.11648/j.sjams.20261404.11

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A Generalized Poisson–Gamma Hybrid Model for Heavy-Tailed Count Data

Isiak Kamaldeen Olomoda^*

, Musa Yunus Olatunji

Published in Science Journal of Applied Mathematics and Statistics (Volume 14, Issue 4)

Received: 2 March 2026 Accepted: 16 March 2026 Published: 27 July 2026

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Abstract

In applications including insurance claim counts, epidemiology, reliability analysis, finance, and network traffic modeling, count data with over dispersion and heavy-tailed behavior are common. Because they impose exponentially decaying tails that underestimate extreme count probabilities, classical Poisson and Poisson–Gamma (Negative Binomial) models are frequently insufficient in such situations. In this study, we propose a generalized Poisson–Gamma hybrid distribution obtained by compounding a Poisson distribution with a power–Gamma mixing law. The proposed model extends the classical Poisson–Gamma framework by introducing an additional shape parameter that governs tail thickness and induces greater dispersion. An explicit infinite-series representation of the probability mass function is derived, and fundamental distributional properties are investigated. It was shown that the Negative Binomial distribution arises as a special case, ensuring model coherence. Furthermore, the proposed model exhibits heavier-than-exponential tails, and under suitable parameter regimes, its tail probabilities display polynomial decay, placing the distribution within the class of heavy-tailed count models. Estimation and inferential aspects were discussed and applications to simulated count data and Monte Carlo simulated count data were also discussed. Heavy-tailed count data demonstrate superior performance compared to classical alternatives.

Published in	Science Journal of Applied Mathematics and Statistics (Volume 14, Issue 4)
DOI	10.11648/j.sjams.20261404.11
Page(s)	90-105
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2026. Published by Science Publishing Group

Keywords

Negative Binomial, Power-Gamma Mixing, Poisson–Gamma hybrid, Heavy-Tailed, Count Data, Classical Poisson, Over – dispersion

References

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APA Style

Olomoda, I. K., Olatunji, M. Y. (2026). A Generalized Poisson–Gamma Hybrid Model for Heavy-Tailed Count Data. Science Journal of Applied Mathematics and Statistics, 14(4), 90-105. https://doi.org/10.11648/j.sjams.20261404.11

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ACS Style

Olomoda, I. K.; Olatunji, M. Y. A Generalized Poisson–Gamma Hybrid Model for Heavy-Tailed Count Data. Sci. J. Appl. Math. Stat. 2026, 14(4), 90-105. doi: 10.11648/j.sjams.20261404.11

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AMA Style

Olomoda IK, Olatunji MY. A Generalized Poisson–Gamma Hybrid Model for Heavy-Tailed Count Data. Sci J Appl Math Stat. 2026;14(4):90-105. doi: 10.11648/j.sjams.20261404.11

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@article{10.11648/j.sjams.20261404.11,
  author = {Isiak Kamaldeen Olomoda and Musa Yunus Olatunji},
  title = {A Generalized Poisson–Gamma Hybrid Model for 
Heavy-Tailed Count Data},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {14},
  number = {4},
  pages = {90-105},
  doi = {10.11648/j.sjams.20261404.11},
  url = {https://doi.org/10.11648/j.sjams.20261404.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20261404.11},
  abstract = {In applications including insurance claim counts, epidemiology, reliability analysis, finance, and network traffic modeling, count data with over dispersion and heavy-tailed behavior are common. Because they impose exponentially decaying tails that underestimate extreme count probabilities, classical Poisson and Poisson–Gamma (Negative Binomial) models are frequently insufficient in such situations. In this study, we propose a generalized Poisson–Gamma hybrid distribution obtained by compounding a Poisson distribution with a power–Gamma mixing law. The proposed model extends the classical Poisson–Gamma framework by introducing an additional shape parameter that governs tail thickness and induces greater dispersion. An explicit infinite-series representation of the probability mass function is derived, and fundamental distributional properties are investigated. It was shown that the Negative Binomial distribution arises as a special case, ensuring model coherence. Furthermore, the proposed model exhibits heavier-than-exponential tails, and under suitable parameter regimes, its tail probabilities display polynomial decay, placing the distribution within the class of heavy-tailed count models. Estimation and inferential aspects were discussed and applications to simulated count data and Monte Carlo simulated count data were also discussed. Heavy-tailed count data demonstrate superior performance compared to classical alternatives.},
 year = {2026}
}

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TY  - JOUR
T1  - A Generalized Poisson–Gamma Hybrid Model for 
Heavy-Tailed Count Data
AU  - Isiak Kamaldeen Olomoda
AU  - Musa Yunus Olatunji
Y1  - 2026/07/27
PY  - 2026
N1  - https://doi.org/10.11648/j.sjams.20261404.11
DO  - 10.11648/j.sjams.20261404.11
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 90
EP  - 105
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20261404.11
AB  - In applications including insurance claim counts, epidemiology, reliability analysis, finance, and network traffic modeling, count data with over dispersion and heavy-tailed behavior are common. Because they impose exponentially decaying tails that underestimate extreme count probabilities, classical Poisson and Poisson–Gamma (Negative Binomial) models are frequently insufficient in such situations. In this study, we propose a generalized Poisson–Gamma hybrid distribution obtained by compounding a Poisson distribution with a power–Gamma mixing law. The proposed model extends the classical Poisson–Gamma framework by introducing an additional shape parameter that governs tail thickness and induces greater dispersion. An explicit infinite-series representation of the probability mass function is derived, and fundamental distributional properties are investigated. It was shown that the Negative Binomial distribution arises as a special case, ensuring model coherence. Furthermore, the proposed model exhibits heavier-than-exponential tails, and under suitable parameter regimes, its tail probabilities display polynomial decay, placing the distribution within the class of heavy-tailed count models. Estimation and inferential aspects were discussed and applications to simulated count data and Monte Carlo simulated count data were also discussed. Heavy-tailed count data demonstrate superior performance compared to classical alternatives.
VL  - 14
IS  - 4
ER  -

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Author Information

Isiak Kamaldeen Olomoda

Department of Mathematics, Nigerian Army University Biu, Biu, Nigeria

Contact Email

http://orcid.org/0009-0000-1767-2709
Musa Yunus Olatunji

Department of Statistics, Kwara State Polytechnic Ilorin, Ilorin, Nigeria

http://orcid.org/0009-0000-0562-0776

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Plain Text BibTeX RIS

APA Style

Olomoda, I. K., Olatunji, M. Y. (2026). A Generalized Poisson–Gamma Hybrid Model for Heavy-Tailed Count Data. Science Journal of Applied Mathematics and Statistics, 14(4), 90-105. https://doi.org/10.11648/j.sjams.20261404.11

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ACS Style

Olomoda, I. K.; Olatunji, M. Y. A Generalized Poisson–Gamma Hybrid Model for Heavy-Tailed Count Data. Sci. J. Appl. Math. Stat. 2026, 14(4), 90-105. doi: 10.11648/j.sjams.20261404.11

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AMA Style

Olomoda IK, Olatunji MY. A Generalized Poisson–Gamma Hybrid Model for Heavy-Tailed Count Data. Sci J Appl Math Stat. 2026;14(4):90-105. doi: 10.11648/j.sjams.20261404.11

Copy | Download

@article{10.11648/j.sjams.20261404.11,
  author = {Isiak Kamaldeen Olomoda and Musa Yunus Olatunji},
  title = {A Generalized Poisson–Gamma Hybrid Model for 
Heavy-Tailed Count Data},
  journal = {Science Journal of Applied Mathematics and Statistics},
  volume = {14},
  number = {4},
  pages = {90-105},
  doi = {10.11648/j.sjams.20261404.11},
  url = {https://doi.org/10.11648/j.sjams.20261404.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.sjams.20261404.11},
  abstract = {In applications including insurance claim counts, epidemiology, reliability analysis, finance, and network traffic modeling, count data with over dispersion and heavy-tailed behavior are common. Because they impose exponentially decaying tails that underestimate extreme count probabilities, classical Poisson and Poisson–Gamma (Negative Binomial) models are frequently insufficient in such situations. In this study, we propose a generalized Poisson–Gamma hybrid distribution obtained by compounding a Poisson distribution with a power–Gamma mixing law. The proposed model extends the classical Poisson–Gamma framework by introducing an additional shape parameter that governs tail thickness and induces greater dispersion. An explicit infinite-series representation of the probability mass function is derived, and fundamental distributional properties are investigated. It was shown that the Negative Binomial distribution arises as a special case, ensuring model coherence. Furthermore, the proposed model exhibits heavier-than-exponential tails, and under suitable parameter regimes, its tail probabilities display polynomial decay, placing the distribution within the class of heavy-tailed count models. Estimation and inferential aspects were discussed and applications to simulated count data and Monte Carlo simulated count data were also discussed. Heavy-tailed count data demonstrate superior performance compared to classical alternatives.},
 year = {2026}
}

Copy | Download

TY  - JOUR
T1  - A Generalized Poisson–Gamma Hybrid Model for 
Heavy-Tailed Count Data
AU  - Isiak Kamaldeen Olomoda
AU  - Musa Yunus Olatunji
Y1  - 2026/07/27
PY  - 2026
N1  - https://doi.org/10.11648/j.sjams.20261404.11
DO  - 10.11648/j.sjams.20261404.11
T2  - Science Journal of Applied Mathematics and Statistics
JF  - Science Journal of Applied Mathematics and Statistics
JO  - Science Journal of Applied Mathematics and Statistics
SP  - 90
EP  - 105
PB  - Science Publishing Group
SN  - 2376-9513
UR  - https://doi.org/10.11648/j.sjams.20261404.11
AB  - In applications including insurance claim counts, epidemiology, reliability analysis, finance, and network traffic modeling, count data with over dispersion and heavy-tailed behavior are common. Because they impose exponentially decaying tails that underestimate extreme count probabilities, classical Poisson and Poisson–Gamma (Negative Binomial) models are frequently insufficient in such situations. In this study, we propose a generalized Poisson–Gamma hybrid distribution obtained by compounding a Poisson distribution with a power–Gamma mixing law. The proposed model extends the classical Poisson–Gamma framework by introducing an additional shape parameter that governs tail thickness and induces greater dispersion. An explicit infinite-series representation of the probability mass function is derived, and fundamental distributional properties are investigated. It was shown that the Negative Binomial distribution arises as a special case, ensuring model coherence. Furthermore, the proposed model exhibits heavier-than-exponential tails, and under suitable parameter regimes, its tail probabilities display polynomial decay, placing the distribution within the class of heavy-tailed count models. Estimation and inferential aspects were discussed and applications to simulated count data and Monte Carlo simulated count data were also discussed. Heavy-tailed count data demonstrate superior performance compared to classical alternatives.
VL  - 14
IS  - 4
ER  -

Copy | Download