This article addresses the challenge of estimating the stress-strength reliability $ \vartheta = \, P[Q < W < Y] $ under the scenario where strength $ (W) $ is in between upper $ (Y) $ and lower $ (Q) $ conditions. The study assumed that these variables were independent and followed the exponentiated Pareto distribution (EPD) with a common unknown shape parameter. The primary goal was to enhance statistical inference for this reliability metric by employing an adaptive progressive Type Ⅱ hybrid censoring scheme (APTII-HCS), a method that ensures the collection of a fixed number of failures while effectively reducing experimental time. Using the maximum likelihood method, we obtained point estimates for the system reliability. Subsequently, the delta method was applied to establish the asymptotic confidence intervals for these estimates. Furthermore, Bayesian estimates were produced under both symmetric (squared error) and asymmetric (linear exponential) loss functions using Markov chain Monte Carlo procedures, which established reliable highest posterior density credible intervals. Extensive simulations were used to assess the effectiveness of Bayesian and classical statistical inference techniques. We conclude that the suggested estimators for the EPD model's reliability metrics and unknown parameters perform well according to several measures of accuracy. The importance of these findings lies in the successful integration of the APTII-HCS framework with the multi-stress-strength model, a combination not previously explored in existing literature. Lastly, three real-world data sets were analyzed to validate the proposed methods. This study provides a robust, flexible methodology for practitioners in engineering, medicine, and environmental science, where maintaining system variables within precise bounds is essential for ensuring durability and safety.
Citation: Ehab M. Almetwally, Amal S. Hassan. Enhanced estimators for the multi-stress strength reliability using an advanced progressive hybrid censoring scheme[J]. AIMS Mathematics, 2026, 11(3): 7740-7765. doi: 10.3934/math.2026318
This article addresses the challenge of estimating the stress-strength reliability $ \vartheta = \, P[Q < W < Y] $ under the scenario where strength $ (W) $ is in between upper $ (Y) $ and lower $ (Q) $ conditions. The study assumed that these variables were independent and followed the exponentiated Pareto distribution (EPD) with a common unknown shape parameter. The primary goal was to enhance statistical inference for this reliability metric by employing an adaptive progressive Type Ⅱ hybrid censoring scheme (APTII-HCS), a method that ensures the collection of a fixed number of failures while effectively reducing experimental time. Using the maximum likelihood method, we obtained point estimates for the system reliability. Subsequently, the delta method was applied to establish the asymptotic confidence intervals for these estimates. Furthermore, Bayesian estimates were produced under both symmetric (squared error) and asymmetric (linear exponential) loss functions using Markov chain Monte Carlo procedures, which established reliable highest posterior density credible intervals. Extensive simulations were used to assess the effectiveness of Bayesian and classical statistical inference techniques. We conclude that the suggested estimators for the EPD model's reliability metrics and unknown parameters perform well according to several measures of accuracy. The importance of these findings lies in the successful integration of the APTII-HCS framework with the multi-stress-strength model, a combination not previously explored in existing literature. Lastly, three real-world data sets were analyzed to validate the proposed methods. This study provides a robust, flexible methodology for practitioners in engineering, medicine, and environmental science, where maintaining system variables within precise bounds is essential for ensuring durability and safety.
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Ehab M. Almetwally, Amal S. Hassan. Enhanced estimators for the multi-stress strength reliability using an advanced progressive hybrid censoring scheme[J]. AIMS Mathematics, 2026, 11(3): 7740-7765. doi: 10.3934/math.2026318
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