Inference for Stress–Strength Reliability Under Unified Hybrid Censoring: A One-Parameter Model with Applications

This paper addresses the estimation of the multi-component stress–strength reliability when both the strength variables and the stress variable follow the one-parameter Garhy distribution. Data are assumed to arise from a unified hybrid censoring scheme, which generalizes both Type-I and Type-II hybrid censoring. A closed-form expression for the reliability parameter Rm,k=P(atleastmof(X1,…,Xk)>Y) is derived, enabling efficient computation. Three estimation procedures are developed: maximum likelihood estimation (MLE), Bayesian inference using Markov chain Monte Carlo (MCMC) with non-informative priors, and the Tierney–Kadane Laplace-type approximation for posterior moments. For each method, we provide complete mathematical derivations, including the likelihood function under unified hybrid censoring, the posterior conditionals, and the asymptotic distribution of the reliability via the Delta method. Furthermore, Bayesian estimation is extended to asymmetric loss functions, and posterior propriety is formally proven. To check the suitability of the proposed methods, a real data application on generator failure times in power systems is presented.