A Log-Exponential Estimator for the Population Mean Using Auxiliary Information: Second-Order Analysis, Theoretical Comparison and Simulation Study

In this paper, we will discuss and compare the Log-Exponential (LE) estimator for the estimation of the finite population mean in simple random sampling without replacement (SRSWOR). The estimator, combines exponential and logarithmic function of the auxiliary-variable ratio through two parameters α and β. The bias and mean square error of the proposed estimators are calculated using a second-order approximation. The optimal values of the parameters, α∗ and β∗, are derived to minimize the mean squared error (MSE). The method provides strict theoretical efficiency bounds compared with several other contemporary estimators. A Monte Carlo simulation study (B = 5,000 replications) is carried out for three correlation values (ρ ∈ {0.3, 0.6, 0.9}) and three sample sizes (n ∈ {50, 100, 200}) , followed by a sensitivity analysis and a comparison of confidence intervals using the bootstrap method. A real-data example uses the Murthy (1967) factory output data. With optimal parameter values, the estimator ˆ¯DLE has the highest percent relative efficiency (PRE) among all estimators, with PRE= 557 at ρ = 0.9 in the simulation study and PRE> 2,000 in the real-data example. An empirical study is included to validate the current research.