Abstract
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.
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Data availability
The data used in this study are publicly available from the books Punjab Development of Statistics, Bureau of Statistics, Government of the Punjab, Lahore, and can be accessed freely online. No individual consent was required, as the data do not contain any personal or identifiable information. The real data are secondary, and their sources are given in the data analysis section, while the simulated data have been generated using R software (latest v. 4.4.0).
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Acknowledgements
The authors would like to thank Princess Nourah bint Abdulrahman University for supporting this work.
Funding
This work was supported by the Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R299), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
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U. D. and H. M. A. contributed to the conceptualization and overall design of the study. The methodology, formal analysis, investigation, validation, and software implementation were carried out jointly by U. D., H. M. A., and H. M. A. Data curation and resource management were handled by U. D. and H. M. A. The original draft of the manuscript was prepared by U. D. and H. M. A., while reviewing and editing were performed collaboratively by U D., H. M. A., and H. M. A. Supervision was provided by H. M. A., and project administration and funding acquisition were managed by H. M. A. All authors have read and approved the final version of the manuscript.
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Daraz, U., Aljohani, H.M. & Alshanbari, H.M. Transformation-based median estimation under skewed–symmetric distributions with long-memory data applications. Sci Rep (2026). https://doi.org/10.1038/s41598-026-47231-0
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DOI: https://doi.org/10.1038/s41598-026-47231-0
