Very Flexible Weibull Reliability Modeling for Shock Environments Using Unified Censoring Plans

The very flexible Weibull (VF-W) distribution is formulated by expressing its cumulative risk function as a logarithmic composite of auxiliary cumulative risks, making the model particularly well-suited for modeling heterogeneous life behaviors. This model admits a remarkably flexible hazard structure, capable of generating monotone increasing, unimodal (increase-then-decrease), and multi-turning-point shapes, thereby capturing complex failure behaviors far beyond those allowed by the classical Weibull distribution. This paper presents a comprehensive inferential study of the VF-W model through the unified progressive hybrid (UPH) censoring framework for modeling shock-type lifetime data. The UPH scheme integrates the advantages of Type-II, generalized hybrid, and progressive hybrid censoring mechanisms into a unified structure that ensures efficiency and adaptability in reliability testing. Classical inference is developed through maximum likelihood estimation with asymptotic interval construction, while Bayesian inference is performed using independent gamma priors and a Markov iterative algorithm. Extensive Monte Carlo experiments are conducted to evaluate the finite-sample performance of both approaches under various censoring intensities, revealing that the Bayesian MCMC-based estimators and their highest posterior density intervals provide superior precision, coverage, and robustness. The proposed VF-W model using UPH-based strategy is further validated through the analysis of a real shocks dataset, where it demonstrates a comparative performance improvement over existing models. The VF-W model exhibits stable parameter estimation under diverse censoring levels, indicating robustness in incomplete-data scenarios. Furthermore, the model maintains analytical tractability, offering closed-form expressions for key reliability measures, which facilitates practical implementation in different scenarios. The results confirm the VFW model’s strong potential as a unifying and computationally stable tool for reliability modeling, particularly in complex engineering and physical systems operating under stochastic shock environments.