Efficient parameter estimation for the Poisson–Ailamujia distribution under ranked set and simple random sampling

This study presents a comparative analysis of parameter estimation methods for the discrete Poisson–Ailamujia distribution, a model suitable for rare-event data that has received limited attention in the statistical literature, under both simple random sampling (SRS) and ranked set sampling (RSS) frameworks. Common estimation techniques, including maximum likelihood (ML), least squares (LS), and weighted least squares (WLS), are assessed through extensive Monte Carlo simulations using three different parameter configurations. To evaluate the accuracy and robustness of the estimators, performance metrics such as mean squared error (MSE), mean relative error (MRE), and bias are employed, enabling a detailed comparison across sampling schemes and parameter settings. The results indicate that the ML method consistently outperforms the other approaches in terms of both bias and MSE for all sampling designs and parameter configurations. Moreover, the RSS method provides greater accuracy and efficiency than SRS across all estimation techniques. Applications to real datasets from domains including education research, epidemiology, and disaster risk analysis align with the simulation findings. Overall, the results suggest that combining the ML approach with RSS yields substantial methodological advantages, particularly when datasets are small or high estimation accuracy is required.