A novel statistical estimator with mathematical and theoretical perspectives: Its utilization in the radiation domains

Proper estimation of population mean introduces a fundamental issue in survey sampling when the sizes of population units vary significantly, and some auxiliary information is known with certainty; in such cases, probability proportional to size (PPS) sampling provides a theoretically robust and practically beneficial sampling scheme. This issue is particularly acute in radiation sciences, where background radiation exhibits significant spatial heterogeneity across different geographical areas, and traditional estimation methods are no longer sufficient. To overcome this limitation, the current research proposes a set of new estimators of the population mean for PPS sampling based on sound mathematical theory and with clear analytical behavior. The parameters of the proposed estimators are flexible and data-driven parameters that dynamically react to the strength and direction of the relationship between the study variable and auxiliary variables - that is, area size and background radiation intensity, thus giving the model the ability to respond to real-world complexity and variability. The theoretical properties of the proposed estimators are analytically discussed. Both the bias and the mean squared error (MSE) have closed-form expressions derived using first-order approximations, providing clear, understandable measures of estimator performance. A comparative study reveals that the suggested estimators consistently outperform classical ratio estimators, regression estimators, and other existing options in the PPS sampling framework, with lower MSE and higher relative efficiency across a wide range of conditions. The theoretical findings are supported by a comprehensive empirical study grounded in actual data from radiation science, and numerical findings indicate significant decreases in MSE and parallel increases in efficiency. These results make the proposed model a powerful, adaptable, and well-performing tool for estimating the population mean in radiological research scenarios, and for direct implementation in environmental radiation surveillance, radiological risk assessment, and evidence-based population-level decision-making.