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.

The probability proportional to size (PPS) sampling method is frequently applied in engineering and industrial surveys when the auxiliary size measures can be obtained, and the population units are substantially heterogeneous. Nevertheless, classical PPS designs estimators are very susceptible to outliers and influential observations, which are common in engineering data because of measurement errors, extreme operating conditions, or structural variability. This research paper presents a powerful estimation method of the finite population mean using PPS sampling on auxiliary information. The proposed estimator has robust influence functions in the design-based PPS framework, enabling one to balance the impacts of extreme sample units by means of controlled weighting without breaking the sampling design on which it relies. Theoretical properties, such as design unbiasedness, consistency, and asymptotic variance of the estimator, are explored and compared with the traditional PPS estimators. Numerical findings show that the robust estimator significantly decreases the mean squared error compared to existing counterparts. The practical usefulness of the suggested methodology is demonstrated on an engineering dataset on the topic of operational performance measurements. Empirical findings prove the effectiveness of the robust estimator in delivering stable and reliable population estimates as opposed to the conventional PPS-based approaches. The results show that the concept of robustness remains critical in survey estimation when applied to engineering problems and can offer a convenient analysis device to analysts who need to work with heterogeneous and contaminated data. The proposed method provides an effective and flexible alternative to strong population estimation where the PPS sampling designs are applicable.