On improving the finite population mean estimation using auxiliary variables: Practical explanations in traditional calligraphy art and radiation fields

This paper aims to enhance the estimation of a finite population mean by capably using auxiliary information under probability proportional to size (PPS). Simple relationships between study and auxiliary variables are usually estimated by traditional estimators, and they may not be sufficient to represent the complexity of real-world data. To overcome this shortcoming, we have proposed a better estimator that incorporates auxiliary variables more flexibly and efficiently thereby increasing precision and reliability. The conceptually suggested methodology is inspired by two opposite and yet organized fields: traditional art of calligraphy and the radiation fields. Precision, balance and proportional flow are critical in calligraphy to create coherent forms, just as auxiliary variables need to be accurately matched with the study variable to enhance estimation accuracy. The better estimator is produced by adjusting the existing ratio type frameworks and incorporating more parameters, which take into consideration variability and correlation among the variables. Its statistical characteristics, such as bias and the mean square error (MSE) are mathematically derived and discussed. Comparisons of efficiencies with the conventional estimators show the proposed method is always better, especially when the auxiliary variables are highly correlated which the variable of interest in the population. The theoretical results are supported by numerical examples and modelling with the help of representative data. It has been verified that the proposed estimator has lower MSE and more efficient across various sampling scenarios. This is particularly handy in applied areas where cost or availability are prohibitive or scarce, and ancillary information is readily accessible. Finally, the research offers an enriched methodology for finite population mean estimation with PPS sampling, both theoretically and practically.