Optimized generalized estimator for valuation of population mean under probability proportional to size sampling using auxiliary variables

The key issue in survey sampling is the efficient estimation of population mean under probability proportional to size sampling (PPS), especially when population units vary considerably in size. In this article, an optimization of generalized estimator of population mean under PPS sampling was proposed by including two auxiliary variables. The proposed class of estimators was designed in such a way that it was flexible in using known population parameters of auxiliary variables, thereby enhancing the efficiency of the estimators. The bias and the mean squared error (MSE) were derived up to the first-order approximation for all the considered estimators. Theoretical comparisons showed that in realistic conditions, the proposed estimator performed better than existing estimators. To validate the theoretical results, an empirical study using real and simulated data was carried out, demonstrating significant improvements in efficiency. The numerical findings showed that the incorporation of auxiliary information in the PPS framework can greatly increase the precision of population mean estimation. The proposed methodology offers a useful contribution to the field of survey sampling and can be effectively implemented, when the population size varies from one individual to another.