Empirical Bayes and Bootstrap Inference for Multicomponent Stress–Strength Reliability Under Ordered Ranked Set Sampling

This paper develops a unified inferential framework for multicomponent stress–strength reliability when the number of active components is stochastic rather than fixed. The system size is modeled by a left-truncated binomial distribution, while strength and stress information are collected through two complementary designs: ordered ranked set sampling and ordered lower record ranked set sampling, respectively. Assuming generalized exponential distributions for strength and stress, we derive the joint likelihood and obtain maximum likelihood, bootstrap, Bayesian, and empirical Bayes estimators of the system reliability. The Bayesian analysis incorporates balanced squared error and balanced linear exponential loss functions, while the empirical Bayes procedure estimates prior hyperparameters directly from the data. Extensive Monte Carlo simulations show that the empirical Bayes estimators, particularly under asymmetric loss, generally achieve lower mean squared errors than their competing procedures, while the bootstrap intervals maintain coverage close to the nominal level. Applications to carbon-fiber strength and annual maximum flood- discharge data demonstrate the practical relevance of the proposed framework. The results establish a flexible reliability methodology that accounts simultaneously for stochastic system size and sampling designs tailored to the distinct nature of strength and stress observations.