Reliability Estimation of s‐Out‐of‐k Systems Under Inverted Z–Lindley Stress–Strength Models With Progressive Censoring: An Engineering Application

ABSTRACT Fault‐tolerant systems are built with redundancy, surviving as long as enough of their components withstand the operating load. We model an ‐out‐of‐ : G system whose nominally identical components share a common random stress and quantify its reliability when both strengths and stress follow inverted Z–Lindley (IZL) laws observed under independent progressive Type–II censoring. Despite the non‐elementary factor , a reciprocal substitution and a binomial expansion reduce the reliability integral to a finite sum of gamma moments, giving a closed form for every , verified to machine precision and shown to decrease in the survival threshold , increase in the redundancy , increase in the strength parameter and decrease in the stress parameter. Treating as design constants, we estimate the two shapes separately and substitute them into the closed form. The IZL law itself admits a closed‐form mode and a full characterization of which fractional moments exist, complementing the divergence of the mean. We compare five frequentist estimators, a Bayesian layer under gamma priors with three loss functions computed by Metropolis–Hastings and by the Tierney–Kadane device, and five interval procedures; full method names are given in the Introduction to keep this summary short. A Monte Carlo study with Monte Carlo standard errors, a prior‐sensitivity analysis, and an application to insulating‐fluid breakdown data—compared against the inverse Lindley and inverse Weibull models by KS, Anderson–Darling and Cramér–von Mises criteria—complete the study, including a genuine progressive Type–II censored analysis of the real data under a disclosed withdrawal scheme, not only of simulated samples.