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

Mohsin Abbas; Muhammad Ahmed Shehzad; Hasnain Iftikhar; Paulo Canas Rodrigues; Abdulmajeed Atiah Alharbi; Jeza Allohibi; Mohsin Abbas; Muhammad Ahmed Shehzad; Hasnain Iftikhar; Paulo Canas Rodrigues; Abdulmajeed Atiah Alharbi; Jeza Allohibi

doi:10.3934/math.20251041

AIMS Mathematics

2025, Volume 10, Issue 10: 23429-23466. doi: 10.3934/math.20251041

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

1.
Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan
2.
Department of Statistics, University of Peshawar, Peshawar 25120, Pakistan
3.
Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan
4.
Department of Statistics, Federal University of Bahia, Salvador 40170-110, Brazil
5.
Department of Mathematics, College of Science, Taibah University, Madinah 42353, Saudi Arabia

Received: 27 July 2025 Revised: 23 September 2025 Accepted: 30 September 2025 Published: 15 October 2025
MSC : 62D05, 62G05, 62H12

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}^2_{Y, DU} $, achieved the highest efficiency ($ RE_3 = 527.69 $), closely followed by the biased difference estimator, $ \hat{S}^2_{Y, DB} $ ($ RE_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.
- multi-stage cluster sampling,
- variance estimation,
- raw moments, auxiliary information,
- difference estimator,
- biased and unbiased estimator,
- relative efficiency,
- absolute bias, survey sampling,
- real-world dataset
Citation: Mohsin Abbas, Muhammad Ahmed Shehzad, Hasnain Iftikhar, Paulo Canas Rodrigues, Abdulmajeed Atiah Alharbi, Jeza Allohibi. Efficient estimators of finite population variance using raw moments under two- and three-stage cluster sampling schemes[J]. AIMS Mathematics, 2025, 10(10): 23429-23466. doi: 10.3934/math.20251041

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Abstract

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}^2_{Y, DU} $, achieved the highest efficiency ($ RE_3 = 527.69 $), closely followed by the biased difference estimator, $ \hat{S}^2_{Y, DB} $ ($ RE_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.

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