A new shifted Lagrangian exponential Poisson model with applications to overdispersed and zero-inflated count data

This paper introduces a new class of discrete distributions, named the shifted Lagrangian exponential Poisson (SLEP) distribution, constructed via a Lagrangian functional equation for the probability-generating function. The resulting model admits an explicit probability mass function derived using Lagrange inversion and is supported on the positive integers. Its structure provides a framework for modeling overdispersed count data, with parameters controlling dispersion and tail behavior. A zero-inflated extension of the SLEP distribution is proposed to model extra zeros, based on a mixture representation of a point mass at zero and the SLEP distribution for positive counts. This formulation enables the simultaneous modeling of zero inflation and overdispersion within a unified framework. A regression extension is further proposed to incorporate covariate effects through log and logit link functions for the positive-count and zero-inflation components, respectively. The model is examined on the DoctorVisits dataset. The empirical results indicate that the zero-inflated SLEP model performs better than the traditional alternatives in log-likelihood, Akaike information criterion, Bayesian information criterion, and chi-square goodness-of-fit measures, including Poisson, negative binomial, and zero-inflated Poisson models. The bootstrap analysis indicates that the estimation procedure is stable, with insignificant bias, accurate standard errors, and satisfactory coverage probabilities. The proposed framework provides a theoretically robust and adaptable solution for modeling intricate count data characterized by overdispersion and zero inflation, and is applicable in medical, actuarial, and social science contexts.