Maximum-Likelihood Estimation for the Zero-Inflated Polynomial-Adjusted Poisson Distribution

Jong-Seung Lee and Hyung-Tae Ha ()
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Jong-Seung Lee: Department of Next Generation Smart Energy System Convergence, Gachon University, 1342 Seongnam-daero, Sujeong-gu, Seongnam-si 13557, Gyeonggi-do, Republic of Korea
Hyung-Tae Ha: Department of Next Generation Smart Energy System Convergence, Gachon University, 1342 Seongnam-daero, Sujeong-gu, Seongnam-si 13557, Gyeonggi-do, Republic of Korea

Mathematics, 2025, vol. 13, issue 15, 1-19

Abstract: We propose the zero-inflated Polynomially Adjusted Poisson (zPAP) model. It extends the usual zero-inflated Poisson by multiplying the Poisson kernel with a nonnegative polynomial, enabling the model to handle extra zeros, overdispersion, skewness, and even multimodal counts. We derive the maximum-likelihood framework—including the log-likelihood and score equations under both general and regression settings—and fit zPAP to the zero-inflated, highly dispersed Fish Catch data as well as a synthetic bimodal mixture. In both cases, zPAP not only outperforms the standard zero-inflated Poisson model but also yields reliable inference via parametric bootstrap confidence intervals. Overall, zPAP is a clear and tractable tool for real-world count data with complex features.

Keywords: zero-inflated poisson; polynomially adjusted poisson; maximum likelihood estimation; multimodality; count regression; fish catch dataset; parametric bootstrapping (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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