Objective Bayesian estimation of time-dependent stress-strength reliability with constant stress and random cycle times

We study time-dependent stress-strength reliability for a system exposed to a constant stress at random cycle times. Initial strength follows an inverse Pareto distribution, cycle arrivals follow a homogeneous Poisson process, and strength decreases by a fixed amount after each cycle. We define failure as an absorbing first-passage event and specify the calibration experiment that supports the likelihood. Maximum likelihood inference is supplemented by an objective Bayesian analysis based on local changes in the cumulative probabilities of the latent strength and interarrival distributions. The resulting improper prior is free of elicited hyperparameters, has a proper posterior whenever at least one cycle is observed, and retains gamma posterior distributions. The posterior mean of the reliability is available as an explicit convergent series. A reproducible simulation study compares point and interval estimators while varying sample size at a fixed time horizon, and a sensitivity study quantifies the effect of misspecifying the known stress and degradation constants. A worked synthetic example illustrates the complete procedure and its diagnostic checks.