Comprehensive Reliability Analysis of the Exponential‐Power Model and Its Applications in Reliability Engineering and Energy Sciences

Reliability analysis in engineering studies and energy‐resource assessments relies on flexible lifetime models and practical censoring schemes that can accommodate heterogeneous failure behaviors and experimental constraints. Motivated by the importance, this study proposes a comprehensive reliability investigation for the exponential‐power (Exp‐P) distribution under the unified Type‐I progressive hybrid censoring (UPH‐T1C) plan, synthesizing advanced statistical theory with practical design considerations. The Exp–P distribution, capable of modeling increasing, decreasing, bathtub‐ and J‐shaped hazard rates, offers a versatile alternative to classical exponential family models. When coupled with UPH‐T1C, which enforces an upper bound on test duration and guarantees a minimum number of observed failures, the resulting likelihood exhibits high nonlinearity, posing considerable inferential challenges. To address this, we develop a unified frequentist–Bayesian estimation paradigm. Maximum likelihood estimators are obtained from nonlinear score equations, and their stability is validated through likelihood surface analysis. In the Bayesian framework, independent gamma priors are combined with a Metropolis–Hastings MCMC algorithm to compute credible and highest posterior density intervals for parameters, reliability functions, and hazard rates. Extensive Monte Carlo simulations reveal that Bayesian estimators, especially under moderately informative priors, exhibit superior efficiency. The analysis also identifies optimal censoring structures that enhance estimation precision. To highlight the versatility of the proposed modeling framework, two contrasting real‐world applications are examined, including a shocks‐before‐failure dataset that captures discrete stress accumulation and progressive degradation in engineering systems, and a global oil reserves dataset that reflects extreme heterogeneity and heavy‐tailed behavior in large‐scale energy resources. Together, these applications demonstrate the model's ability to accommodate fundamentally different failure‐generating mechanisms, one driven by cumulative shocks and the other by pronounced cross‐sectional variability, within a unified and flexible inferential structure. Applications to cumulative shock data in reliability engineering and global crude oil reserve distributions demonstrate the robustness and interdisciplinary relevance of the proposed framework, bridging engineering reliability analysis with large‐scale economic resource assessments.