New efficient calibrated estimators for the population distribution function using two auxiliary variables

The population distribution function (DF) plays a crucial role in public health; its applications include modeling drug absorption/clearance rates, guiding personalized dosage decisions, and understanding patterns of infectious period durations. It is also used to quantify the fraction of the population exceeding dangerously high exposure or emergency disease markers. Beyond public health, the DF is widely used in geography, environmental science, and agriculture. It supports soil-quality assessment across regions and informs environmental standards for contaminants such as arsenic in groundwater and particulate matter like PM2.5. This study proposes novel calibrated estimators for the population distribution function using two auxiliary variables under simple random sampling. We derive optimal calibration weights for various practical scenarios and formulate the corresponding estimators. A numerical study using two real datasets was conducted, and the findings are further supported through a simulation study. The outcomes are compared with unbiased, ratio, and regression estimators using both single and two auxiliary variables. We observed that the performance of the proposed estimators is superior, demonstrating lower absolute bias (ABS), lower mean squared error (MSE), and higher percent relative efficiency (PRE). The results highlight the effectiveness of the proposed estimators and encourage survey practitioners to apply them in real-world scenarios.