A new generalization of the zero-truncated negative binomial distribution by a Lagrange expansion with associated regression model and applications

In this paper, we propose a new generalization of the zero-truncated negative binomial distribution using the Lagrange expansion of the second kind, which we call the Lagrangian zero-truncated negative binomial distribution (LZTNBD). The proposed distribution’s formulation and properties, including its mean, variance, skewness, kurtosis, factorial moment, and index of dispersion, are studied. Because of the various shapes of the hazard rate function, the new distribution reveals great flexibility. Using the equivalence theorem of the class of Lagrangian distribution, we show that the LZTNBD belongs to the Lagrangian family of the first kind. We estimate the unknown parameters of the LZTNBD using the method of maximum likelihood. The performance of the estimates is evaluated through a broad simulation study. We employ the mean-parameterized form of the LZTNBD to present a novel count regression model that is apt for both overdispersed and underdispersed situations. To further illustrate the relevance and applicability of the proposed model, real-world data sets are employed.