An Enhanced Product Estimator in Two-stage Sampling for Ancillary Characteristic-Based Population Mean Estimation

In this research, a refined two-stage sampling product estimator for estimating the population mean is introduced, incorporating information from auxiliary attributes. The estimation is carried out in a product form within the context of two-stage sampling. The modified product estimator proposed here is a versatile family of estimators, yielding different results depending on the value of Lambda. Specifically, for the projected estimator, when lambda takes values of 0, 0.5, and 1, the product estimators align with three different estimators respectively. When the auxiliary attribute is treated as a variable, the estimator corresponds to another estimator. In the scenario where lambda is 0 and the auxiliary attribute is a variable, the result is akin to ordinary double sampling. The text further concentrates on the derivation of expressions for Bias and Mean Squared Error of the projected modified product estimator up to the first order of approximation. Through a comprehensive comparison with existing related estimators, both theoretically and empirically, it is demonstrated that the proposed modified product estimator exhibits superior efficiency at the optimal value of Lambda. This research contributes to the field by showcasing the enhanced proficiency of the proposed estimator in comparison to its counterparts.