On estimation $R=P(X>Y)$ under classical and median ranked set sampling based on inverse Rayleigh distribution

In this effort, we estimate $R = P(X > Y)$ under the classical and median ranked set sampling schemes, assuming that $X$ and $Y$ follow the inverse Rayleigh distribution. To address this, we derive the maximum likelihood estimator of $R$ using iterative algorithms due to the lack of a closed-form expression. We also employ a modified maximum likelihood approach as an alternative, allowing a closed-form estimator for $R$. A simulation study compares the performance of the $R$ estimators under both classical and median ranked set sampling schemes and simple random sampling, including an evaluation of asymptotic confidence intervals. Furthermore, acknowledging that perfect ranking is often unrealistic, we extend the study by incorporating an imperfect ranking model based on probabilistic misclassification. Finally, two real data examples are presented to illustrate and support the simulation results.