The Neutrosophic Zero-Inflated Poisson Distribution: Properties and Applications in Imprecise Count Data

The Zero-Inflated Poisson (ZIP) model is a cornerstone for analyzing count data characterized by an excess of zero observations. However, classical statistical models often fail when data is sourced from complex, vague, or indeterminate environments. Building upon recent advances in neutrosophic statistics, this paper introduces the Neutrosophic Zero-Inflated Poisson (NZIP) distribution, a generalization of the ZIP model within the framework of neutrosophic probability. By treating the mixing probability and/or the Poisson rate parameter as neutrosophic numbers, the proposed model can effectively handle uncertainty and indeterminacy inherent in the data-generating process. We derive key mathematical properties of the NZIP, including its probability mass function, moments, moment-generating function, and index of dispersion. The parameters are estimated using the method of Neutrosophic Maximum Likelihood Estimation (NMLE); we show that only the neutrosophic parameters ω_N and λ_N (not their determinate and indeterminate parts separately) are identifiable from count data, and we construct their indeterminacy intervals from the data as profile-likelihood regions. A reproducible simulation study and two illustrative applications to healthcare utilization and manufacturing defect frequency data show that, at their best point, the NZIP models coincide with the classical ZIP fit, while adding interval bands for the parameters, the expected frequencies and the reliability and risk measures. The proposed distribution is a tool for imprecise count data in which this indeterminacy must be reported explicitly.