Maximum-Likelihood Estimation for the Zero-Inflated Polynomial-Adjusted Poisson Distribution
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-l...
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2025-07-01
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| author | Jong-Seung Lee Hyung-Tae Ha |
| author_facet | Jong-Seung Lee Hyung-Tae Ha |
| author_sort | Jong-Seung Lee |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-38c0551c6ebd48c9b9e19bd040bfa483 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-07-01 |
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| series | Mathematics |
| spelling | doaj-art-38c0551c6ebd48c9b9e19bd040bfa4832025-08-20T04:00:50ZengMDPI AGMathematics2227-73902025-07-011315238310.3390/math13152383Maximum-Likelihood Estimation for the Zero-Inflated Polynomial-Adjusted Poisson DistributionJong-Seung Lee0Hyung-Tae Ha1Department of Next Generation Smart Energy System Convergence, Gachon University, 1342 Seongnam-daero, Sujeong-gu, Seongnam-si 13557, Gyeonggi-do, Republic of KoreaDepartment of Next Generation Smart Energy System Convergence, Gachon University, 1342 Seongnam-daero, Sujeong-gu, Seongnam-si 13557, Gyeonggi-do, Republic of KoreaWe 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.https://www.mdpi.com/2227-7390/13/15/2383zero-inflated poissonpolynomially adjusted poissonmaximum likelihood estimationmultimodalitycount regressionfish catch dataset |
| spellingShingle | Jong-Seung Lee Hyung-Tae Ha Maximum-Likelihood Estimation for the Zero-Inflated Polynomial-Adjusted Poisson Distribution Mathematics zero-inflated poisson polynomially adjusted poisson maximum likelihood estimation multimodality count regression fish catch dataset |
| title | Maximum-Likelihood Estimation for the Zero-Inflated Polynomial-Adjusted Poisson Distribution |
| title_full | Maximum-Likelihood Estimation for the Zero-Inflated Polynomial-Adjusted Poisson Distribution |
| title_fullStr | Maximum-Likelihood Estimation for the Zero-Inflated Polynomial-Adjusted Poisson Distribution |
| title_full_unstemmed | Maximum-Likelihood Estimation for the Zero-Inflated Polynomial-Adjusted Poisson Distribution |
| title_short | Maximum-Likelihood Estimation for the Zero-Inflated Polynomial-Adjusted Poisson Distribution |
| title_sort | maximum likelihood estimation for the zero inflated polynomial adjusted poisson distribution |
| topic | zero-inflated poisson polynomially adjusted poisson maximum likelihood estimation multimodality count regression fish catch dataset |
| url | https://www.mdpi.com/2227-7390/13/15/2383 |
| work_keys_str_mv | AT jongseunglee maximumlikelihoodestimationforthezeroinflatedpolynomialadjustedpoissondistribution AT hyungtaeha maximumlikelihoodestimationforthezeroinflatedpolynomialadjustedpoissondistribution |