Efficient estimators of finite population variance using raw moments under two- and three-stage cluster sampling schemes

In this study, we proposed novel estimators for finite population variance based on the raw moments of the study and auxiliary variables. Specifically, we developed both biased and unbiased estimators of variance using the raw moments of the study variable alone, as well as biased and unbiased difference-type estimators that incorporate the raw moments of a single auxiliary variable. These estimators were evaluated under two-stage cluster sampling (2SCS) and three-stage cluster sampling (3SCS) schemes. Their performance, with and without auxiliary information, was assessed using mean squared error (MSE), absolute bias (AB), and relative efficiency (RE) criteria. Results from two real populations showed that AB decreases and RE improves with increasing sample size. Notably, under 3SCS, the unbiased difference estimator, ${\hat{S}}_{Y,DU}^2$, achieved the highest efficiency ( $R{E}_3=527.69$), closely followed by the biased difference estimator, ${\hat{S}}_{Y,DB}^2$ ( $R{E}_4=527.26$). Both estimators substantially outperformed conventional variance estimators without auxiliary information (baseline $RE$ = 100). These findings demonstrate that incorporating auxiliary variables significantly enhances estimation accuracy, offering a practical and robust approach for variance estimation in complex survey designs.