A Unifying framework for exponential–Gamma related lifetime distributions with special cases and extensions

Abstract

Lifetime and reliability modeling has motivated the development of flexible probability distributions capable of capturing diverse hazard rate behaviors. Two research streams have dominated the literature: parameter-driven generalizations of the Gamma distribution and mixture- or transformation-based Exponential–Gamma constructions. However, the lack of a unified framework limits systematic comparison and further extensions. This study presents a coherent framework that integrates these approaches, unifying classical and higher-order Exponential–Gamma distributions. The framework employs finite mixture constructions, product-type mechanisms, and transformation operators such as exponentiation and generalized exponentiation, yielding models including the Exponential, Gamma, Lindley, New Exponential–Gamma, Exponentiated New Exponential–Gamma, and Exponentiated Generalized New Exponential–Gamma distributions as special cases or extensions. Parameters were estimated using the Maximum Likelihood Method, and the models were applied to real-life lifetime data. Empirical evaluation based on −2 log-likelihood, AIC, and BIC demonstrates that higher-order Exponential–Gamma models outperform classical and related competing distributions. Key properties are derived in closed form, and graphical and numerical illustrations confirm the nesting structure and progressive increase in flexibility. These results highlight the practical usefulness of the framework and establish it as a coherent theoretical foundation for existing models, providing a flexible platform for developing new lifetime distributions.