A Modified Log-Exponential Estimator for Finite Population Mean Using Skewness and Kurtosis

In this article, a Modified Log-Exponential (MLE) estimator is proposed, which is an extension of the original Log-Exponential (LE) estimator for estimating the mean of a finite population. The main contribution of this article is that it uses the skewness (γ1) and kurtosis (β2) of the auxiliary variable as automatic calibration weights for the power and logarithmic parameters, which adapt autonomously to the characteristics of the population, without compromising the optimality of the classical LE estimator. To rigorously evaluate the proposed estimator, I derive closed-form expressions for the Bias and Mean Square Error (MSE) of both the LE and MLE estimators, and establish formal efficiency theorems to benchmark the MLE estimator against the LE estimator and thirteen other popular estimators. I further conduct a Monte Carlo simulation study, consisting of B = 5,000 replications, under four different population distributions (normal, lognormal, skewed, heavy-tailed), four different correlation values (ρ ∈ {0.3,0.5,0.7,0.9}), and three different sample sizes (n ∈ {50,100,200}). To further verify the results, I also perform bootstrap 95% confidence interval width comparison and validation of results using two commonly used benchmark datasets. The results show that the MLE estimator maintains the efficiency of the LE estimator for symmetric populations, while achieving much smaller MSE values and robustness for skewed or heavy-tailed data. Hence, the proposed estimator appears to be a better alternative for use in a variety of estimation scenarios in practice.