MSE comparison of estimators.

Two-stage cluster sampling is often employed in survey sampling when complete population information is not available. In this setting, the Exponentially Weighted Moving Average (EWMA) statistic offers an efficient way to estimate the population mean by incorporating both past and current data. Motivated by this, we propose a class of memory-type ratio and exponential estimators for estimating the population mean under a two-stage cluster sampling framework. Theoretical expressions for the biases and mean square errors (MSE) of the proposed estimators are derived. To evaluate their performance, a comprehensive simulation study was carried out, supplemented by an empirical application. Several special cases of the proposed estimators were also considered and compared with existing two-stage estimators. The analysis was performed under different values of the EWMA smoothing constant (). Both simulation and empirical results consistently show that the proposed memory-type two-stage ratio estimators outperform existing approaches, providing improved efficiency with minimum MSE.