Estimation of the generalized stress-strength reliability for the proportional hazards family

The stress-strength reliability measure, traditionally defined as R = P(Y<X), quantifies the probability that a system with random strength X can withstand a random stress Y. However, in many real-world scenarios, systems often involve multiple components with varying strengths subjected to multiple stress conditions, requiring a more general framework. A natural generalization arises by considering the probability that a certain order statistic of the stress variables is less than a particular order statistic of the strength variables. This paper develops a comprehensive framework for estimating generalized stress-strength reliability within the proportional hazards family. We develop both classical and Bayesian estimation methods for this measure, analyse their statistical properties, and assess their performance through simulation studies. Additionally, we provide a real-data illustration to demonstrate the practical utility of the proposed methodology.