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

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.