Stress-Strength Reliability for the Poisson–Lindley Distribution Based on the Rank Set Sample Method

The estimation of stress-strength reliability R=PX<Y is a common problem in engineering and statistics. Traditional approaches typically rely on simple random sampling to estimate system reliability. In recent years, however, ranked set sampling has emerged as a cost-effective and efficient alternative to simple random sampling. Despite its growing popularity, the application of ranked set sampling to stress-strength reliability in discrete models remains underexplored in the literature. This paper addresses this gap by presenting a detailed statistical analysis of stress-strength reliability when stress and strength are modeled as independent discrete random variables having Poisson–Lindley distributions. We employ both point estimation and bootstrap confidence interval methods to derive stress-strength reliability estimates under both simple random sampling and ranked set sampling frameworks. To evaluate the effectiveness of these estimates, we conduct extensive Monte Carlo simulations, comparing their performance in different settings. The simulation results, complemented by analyses of two real datasets, show that ranked set sampling estimates generally outperform traditional simple random sampling estimates in terms of efficiency and accuracy. These results highlight the potential advantages of ranked set sampling in estimating system reliability and demonstrate the applicability of the method to discrete models, providing a valuable contribution to the field. The ranked set sampling method provides more consistent and reliable estimates compared to the simple random sampling method, with shorter confidence intervals and lower error rates, especially for small sample sizes.