The estimator development process is more efficient when additional information is used. However, occasionally, it is necessary to use information regarding unknown population parameters. In these cases, we chose two-phase sampling by substituting the population mean of the supplemental variable with the sample mean from first-phase sampling. The goal of this project was to develop effective estimators of the finite population mean in a two-phase sampling scenario with a single auxiliary variable. Under certain conditions, the recommended estimators outperform the current estimators, producing biased and Mean Square Error (MSE) expressions. Empirical and theoretical comparisons of the proposed families were conducted using real and simulated data. We found that the proposed families were more effective in the two-phase sampling situation than in all-population mean estimators.

Unbiased estimators are valuable when no auxiliary information is available beyond the primary study variables. However, once auxiliary information is accessible, biased estimators with smaller Mean Square Error (MSE) often outperform unbiased estimators that have large variances. We sought to develop new estimators that incorporate a single auxiliary variable in stratified random sampling. This study contributes to the field by introducing two distinct families of estimators designed to estimate the finite population mean. We conducted a theoretical evaluation of the estimators' performance by examining bias and MSE derived under first-order approximation. Additionally, we established the theoretical conditions necessary for the proposed estimator families to exhibit superior performance compared with existing alternatives. Empirical and simulation-based studies demonstrated significant improvements in estimators over competing estimators for finite-population parameter estimation.