(untitled) (Statistics, Optimization & Information Computing, 2025)
Severe operational conditions frequently lead to system failure. One frequent error, though, is that systems can quickly become unstable and stop functioning as intended when operating at extremely high or low levels. This article addresses reliability estimation (R = P(Y < X < Z)), emphasizing the constraint that the strength (X) must exceed the lower stress (Y) while remaining below the upper stress (Z). Assuming independent Weibull distributions for strength and stresses, reliability estimation of (R = P(Y < X < Z)\) from frequentist and Bayesian perspectives utilizing upper record values is investigated. This study develops Bayesian estimators for the reliability (R) using three different loss functions: quadratic, linear exponential, and minimum expected. Independent informative (gamma) and non-informative (uniform) priors are assumed, and the corresponding loss functions are incorporated to derive posterior estimates. Bayesian inference for the reliability parameter (R) is performed using Metropolis–Hastings within a Markov Chain Monte Carlo (MCMC) framework. Further, a detailed simulation study to evaluate the performance of the proposed estimators with MCMC techniques is conducted to facilitate the computation of the posterior estimates. Lastly, to validate the proposed methodologies, the reliability estimates are applied to three real jute fiber datasets. Jute fiber, a biodegradable and cost-effective natural material, is examined for its potential in textile applications. The results highlight its favorable mechanical and thermal properties, indicating that jute fiber is a sustainable and efficient alternative for eco-friendly textile materials.