A new Weibull reliability modeling under adaptive progressive censoring and its applications in engineering and physical sciences

This study developed several inferential procedures for a modified and highly flexible extension of the classical Weibull distribution, termed the very flexible Weibull (VFW) distribution. Statistical inference was conducted under an adaptive progressive censoring scheme, which enhances dynamic control over the test duration and sample utilization. Both likelihood-based and Bayesian estimation methods were established for the model parameters and associated reliability measures. Nonlinear likelihood equations were derived and solved numerically using the Newton–Raphson iterative method, and asymptotic confidence intervals were constructed under normal and log-normal approximations. Within the Bayesian framework, independent gamma priors were assumed, and posterior inference was carried out using Markov chain Monte Carlo simulation based on the Metropolis–Hastings algorithm to obtain posterior summaries and two types of credible intervals. A numerical assessment was conducted to evaluate the statistical performance of the proposed estimators under various adaptive censoring configurations and prior specifications. The simulation results demonstrated that Bayesian estimators, particularly under informative priors, provide superior bias reduction, improved coverage probabilities, and more stable interval estimates than their frequentist counterparts. To illustrate the practical applicability of the proposed methodology, two real engineering datasets were analyzed. The data analysis confirmed the excellent fitting capability of the VFW distribution for modeling complex lifetime behaviors. From an engineering standpoint, the data-driven outcomes underscore the VFW distribution as a flexible and dependable lifetime model that facilitates precise reliability evaluation and decision-making in actual industrial systems.