Log-ratio type estimation for the finite population mean under simple random sampling without replacement with theory, simulation and application

Abstract

Abstract We propose two novel logarithmic ratio–type estimators for the finite-population mean under simple random sampling without replacement (SRSWOR). The estimators integrate a logarithmic transformation of the auxiliary variable to stabilize variance, reduce the influence of outliers, and better capture nonlinear relationships between study and auxiliary variables. We derive closed-form expressions for first-order bias and mean squared error (MSE) and obtain analytic expressions for the optimal tuning constants by direct minimization of the approximate MSE. A comprehensive numerical study, comprising five real engineering datasets and extensive Monte-Carlo simulations from multivariate normal, log-normal and gamma populations, evaluates finite-sample behavior across a range of sample sizes and correlation structures. The proposed estimators consistently reduce MSE and deliver large percent-relative-efficiency (PRE) gains relative to the classical sample mean and common competitors (empirical PREs ≈ 283; simulation PREs up to ≈ 670), with especially large and stable improvements under skewed or heavy-tailed populations. Theoretical formulas and simulation evidence align closely, showing robustness to nonlinearity and skewness while retaining simple implementation for practitioners. Results are derived under SRSWOR using first-order approximations; extensions to higher-order corrections, stratified and two-phase designs, and uncertainty in auxiliary means are recommended for future work.