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<p>The Neutrosophic Negative Binomial (NNB) distribution has emerged as a powerful tool for modeling over-dispersed count data imbued with indeterminacy. However, its application has been restricted to cross-sectional settings, limiting its utility for modern complex data structures. This paper presents two significant methodological extensions to bridge this gap. First, we introduce the Neutrosophic Negative Binomial Mixed-Effects Model (NNB-MM) for analyzing clustered or longitudinal count data where the randomness and indeterminacy coexist across subjects or clusters. Second, we propose a Neutrosophic Integer-Valued Autoregressive model of order p with Negative Binomial innovations (NNB-INAR(<i>p</i>)) for time series of counts governed by a higher-order indeterminate process. We provide the rigorous mathematical formulations for both models, introducing the novel concept of a neutrosophic binomial thinning operator and a corresponding neutrosophic Yule-Walker equation system for the NNB-INAR(<i>p</i>). A custom estimation algorithm, based on the principles of neutrosophic statistics and interval optimization, is developed. Simulation studies demonstrate the models’ ability to accurately recover parameters and their superiority over classical models in the presence of indeterminacy. Finally, we illustrate the practical application of the NNB-MM on a longitudinal clinical trial dataset with indeterminate patient reporting and the NNB-INAR(<i>p</i>) on an economic time series of small business failures during a period of high economic uncertainty. The results confirm that the proposed models provide more realistic and robust inferences by formally accounting for the inherent indeterminacy in the data-generating process.</p>

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A Novel Fusion for Uncertain Dynamic Count Data: Neutrosophic Negative Binomial Mixed-Effects and INAR(p) Models

  • 期刊论文

  • 发表日期：

    2026年5月20日

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  • Ibrahim Sadok

摘要

The Neutrosophic Negative Binomial (NNB) distribution has emerged as a powerful tool for modeling over-dispersed count data imbued with indeterminacy. However, its application has been restricted to cross-sectional settings, limiting its utility for modern complex data structures. This paper presents two significant methodological extensions to bridge this gap. First, we introduce the Neutrosophic Negative Binomial Mixed-Effects Model (NNB-MM) for analyzing clustered or longitudinal count data where the randomness and indeterminacy coexist across subjects or clusters. Second, we propose a Neutrosophic Integer-Valued Autoregressive model of order p with Negative Binomial innovations (NNB-INAR(p)) for time series of counts governed by a higher-order indeterminate process. We provide the rigorous mathematical formulations for both models, introducing the novel concept of a neutrosophic binomial thinning operator and a corresponding neutrosophic Yule-Walker equation system for the NNB-INAR(p). A custom estimation algorithm, based on the principles of neutrosophic statistics and interval optimization, is developed. Simulation studies demonstrate the models’ ability to accurately recover parameters and their superiority over classical models in the presence of indeterminacy. Finally, we illustrate the practical application of the NNB-MM on a longitudinal clinical trial dataset with indeterminate patient reporting and the NNB-INAR(p) on an economic time series of small business failures during a period of high economic uncertainty. The results confirm that the proposed models provide more realistic and robust inferences by formally accounting for the inherent indeterminacy in the data-generating process.