Regression-type calibration estimators for efficient design-based finite population estimation

A central challenge in sampling theory is the efficient estimation of finite-population parameters while preserving design-based validity, particularly when auxiliary information is available but not fully exploited by classical design-unbiased estimators. This study develops and evaluates transformation-based regression-type calibration estimators within a design-based inference framework. Specifically, the Linear Regression Model (LRM), Second-Order Polynomial Model (SOPM), and Inverse Term Regression Model (ITRM) are examined using original and transformed auxiliary variables. Using COVID-19 mortality data from Gulf Cooperation Council (GCC) countries as an illustrative finite-population application, their empirical design-based variances and relative efficiencies are compared with the Simple Random Sampling Without Replacement (SRSWOR) estimator. The results show that the regression-type calibration estimators generally achieve lower design-based variance than SRSWOR, with the magnitude of the efficiency gain determined by the finite-population covariance structure linking the study variable to the original and transformed auxiliary variables. Nonlinear transformations provide additional gains when they capture systematic auxiliary information beyond the linear calibration space, subject to adequate numerical stability. Overall, the findings show that transformed auxiliary-variable calibration can yield meaningful finite-sample efficiency gains while retaining a design-based interpretation of survey inference. The GCC application serves solely to illustrate estimator behavior under finite-population sampling.