Transformation-based median estimation under skewed–symmetric distributions with long-memory data applications

In this article, an enhanced class of median estimators for finite populations is formulated within a double-sampling framework. The suggested estimators use transformation-based methods that make the best use of limited extra information, which lowers the cost of collecting data while increasing the accuracy of the estimates. First-order approximations give us analytical equations for bias and mean squared error. To assess performance, comprehensive Monte Carlo simulations carried out utilizing three actual-life data sets and five skewed symmetry probability distributions across various parameter conditions. We investigated the robustness of the estimators even more by using fractional Gaussian noise (fGn) and ARFIMA models with the Hurst exponent, which show long-memory or fractal behavior in auxiliary variables. The new estimators provide more efficient performance than existing methods in terms of accuracy, as evidenced by results based on percent relative efficiency (PRE), with only a slight decrease in efficiency as long-range dependence increases. Graphical analyses validate their reliability, even when supplementary information shifts from short-memory assumptions, confirming that the new transformation-based estimators offer stable and economical techniques for median estimation in two-phase sampling designs.