Mixed-Record Stress-Strength Reliability for the Poisson-Exponential Distribution

Abstract

This study develops classical and Bayesian inference for the stress–strength reliability parameter R=P(X>Y) under an asymmetric mixed-record sampling scheme for the Poisson-exponential distribution. The strength variable is observed through upper record values, whereas the stress variable is represented by lower record values. Exact likelihood contributions are derived from the two record processes and combined under independence. To improve numerical stability when only short record sequences are available, the main inferential formulation assumes a commonly known or externally calibrated rate parameter, while the two shape parameters are estimated from their corresponding records. Maximum likelihood estimation is developed together with a logit-based asymptotic confidence interval using the observed information and the delta method. Bayesian inference is formulated under independent flat working priors, and a parametric bootstrap-Bayesian percentile interval is also considered. A Monte Carlo study evaluates bias, RMSE, empirical coverage probability, and average interval length under low-, moderate-, and high-reliability scenarios. The Bayesian posterior-mean estimator yields lower RMSE than the MLE across all reported settings, while Bayesian intervals generally provide coverage closer to the nominal level and shorter average lengths. The proposed methodology is further illustrated using published steel-specimen lifetime data, demonstrating both the feasibility and uncertainty of reliability inference based on limited mixed-record observations.