A Novel Family of CDF Estimators Under PPS Sampling: Computational, Theoretical, and Applied Perspectives

Accurate estimation of population distribution characteristics is a fundamental task in survey sampling and statistical inference. This paper introduces a new family of estimators for the cumulative distribution function (CDF) under probability proportional to size (PPS) sampling, incorporating auxiliary information to enhance efficiency. The proposed approach employs dual auxiliary variables in the estimation phase, while the sampling design relies on a single auxiliary variable. Theoretical properties, including bias and mean squared error (MSE), are rigorously derived to establish the efficiency of the new class. An extensive empirical evaluation using three distinct populations—fisheries data, wine chemistry data, and demographic records—demonstrates the superiority of the proposed estimators. In terms of accuracy, the best-performing proposed estimator achieves an MSE of 0.0012, compared to 0.0127 for the widely used GK estimator. Percentage relative efficiency (PRE) values further underscore these improvements, with gains ranging from 123% to over 328% across the three populations. Graphical comparisons confirm these trends, illustrating that the proposed estimators consistently dominate conventional approaches. Overall, the findings highlight both the theoretical soundness and practical utility of the proposed family, offering robust and computationally efficient improvements for CDF estimation in complex survey designs.