Theory-Driven Inference and Data Modeling for the Odd Weibull-G Negative Binomial Class

Abstract

The need for flexible lifetime models that accommodate non-monotone hazard shapes has driven the development of numerous generators that extend classical baseline distributions. This paper introduces the odd Weibull--G negative binomial (WG--NB) family, a new class of distributions constructed by compounding the odd Weibull--G generator with a negative binomial scheme. The proposed family is motivated by the limitation of existing generators in capturing bathtub, unimodal, and reversed-J-shaped hazard rates while retaining a tractable analytical structure. Several structural and mathematical properties are derived, including a useful series expansion of the density function, ordinary and incomplete moments, moment generating function, mean deviations, Lorenz and Bonferroni curves, moments of residual and reversed residual life, and Shannon and R\'enyi entropies. A characterization based on two truncated moments is also established. For estimation, the maximum likelihood approach is developed, along with the maximum product spacing and Bayesian approaches. A simulation study examines the performance of the proposed estimators, and the flexibility of the new family is illustrated through the analysis of two real datasets using the AIC, CAIC, HQIC, the Kolmogorov-Smirnov statistic with its P-value, and the Anderson-Darling and Cramer-von Mises statistics. The results show that the WG--NB family provides a better fit than several competing models and offers a practical alternative for modelling lifetime and reliability data.