Classical and Bayesian Inference for Stress-Strength Reliabilityof Shifted Geometric Lifetimes under Unified Hybrid Censoring

Abstract

Stress-strength reliability inference is developed for independent shifted geometric strength and stress lifetimes observed under unified hybrid censoring schemes (UHCSs). The reliability R=P(X > Y) is appropriate for experiments in which lifetime is recorded as a number of operating cycles, inspections, or trials. Under the UHCS the likelihood reduces to a kernel in the number of observed failures and the total discrete exposure, which yields closed-form maximum likelihood estimators of the geometric parameters and, by invariance, of ????. Wald and delta-method intervals are derived from the observed Fisher information, a conservative conditional beta interval is obtained from beta tails with a Bonferroni adjustment, and an exact procedure based on inversion of the UHCS sampling distribution is outlined. Bayesian inference under the Jeffreys prior of the shifted geometric model and under the uniform prior leads to beta posteriors for the parameters and to Monte Carlo evaluation of the posterior of R. A parametric bootstrap percentile interval is also considered. A pilot Monte Carlo study with PX=0.05, PY =0.10, , and sample sizes between 250 and 600 compares bias, mean squared error, coverage probability, and average interval length. The maximum likelihood and Jeffreys estimators are nearly indistinguishable, and the uniform prior adds a small bias. The conditional beta interval attains coverage of at least 0.995 with intervals about 1.6 times longer, whereas the delta-method, Jeffreys credible, and bootstrap intervals have coverage within Monte Carlo error of 0.95. Accelerated capacitor lifetimes converted to inspection cycles illustrate the methods. The estimate of R remains between 0.656 and 0.677 as the cycle width varies from 1 to 200 hours, while the per-cycle parameters change by a factor of about 180.