Estimating the stress–strength reliability parameter \(R=P(X>Y)\) is of great importance in survival and reliability analysis. However, most existing studies focus on specific distributions rather than a general family, and few consider cost-efficient sampling designs such as ranked set sampling. In this endeavour, we investigate the estimation of \(R=P\left( X>Y \right) \) when the strength (X) and stress (Y) variables follow the distribution from the Lehmann family. In this regard, we use four different sampling designs including simple random sampling, ranked set sampling, median ranked set sampling, and extreme ranked set sampling. The estimation of R is obtained in both cases when the common parameter of the baseline cumulative density function ( \(\theta \) ) is unknown and known. The parameter R is estimated by the maximum likelihood method and using the numerical technique for all different sampling schemes and in both cases, except for the design of simple random sampling in the known case of \(\theta \) , where there is a closed form for R. To investigate the performance of different sampling strategies in estimating \( R \) , a Monte Carlo simulation is performed. All results are presented as an example of the Lehmann family using the exponentiated Burr–Hatke distribution. Finally, a real example is used in the field of medicine to explain the application of the studied methods and strengthen the simulation results.
