Optimal Transformation-Based Median Estimation Under Stratified Double Sampling with Limited Auxiliary Information

This study develops a new class of transformation-based estimators for estimating the population median within a stratified two-phase sampling framework. The proposed approach is designed to improve estimation accuracy while reducing survey costs, particularly in situations where auxiliary information is partially available or expensive to collect. By using suitable transformations of the auxiliary variable, the estimators achieve greater stability and robustness in the presence of skewed distributions and extreme observations. Theoretical properties of the proposed estimators are established using first-order approximations, leading to explicit expressions for bias and mean squared error. Optimal conditions are also derived to ensure improved efficiency. To evaluate performance, both simulation experiments and real-world datasets are considered under a variety of distributional settings and correlation structures. The results consistently show that the proposed estimators outperform conventional approaches, including ratio, regression, and exponential-type estimators, in terms of efficiency. In particular, notable improvements are observed for skewed and heavy-tailed populations, where traditional methods often perform poorly. These findings highlight the practical usefulness of the proposed methodology for survey applications in fields such as economics, public health, and social sciences, where reliable median estimation is essential.