Symmetric hybrid transformations for robust mean estimation under stratified systematic sampling

Fatimah A. Almulhim; Umer Daraz; Hassan M. Aljohani; Javid Shabbir; Fatimah A. Almulhim; Umer Daraz; Hassan M. Aljohani; Javid Shabbir

doi:10.3934/math.20261260

AIMS Mathematics

2026, Volume 11, Issue 9: 32046-32080. doi: 10.3934/math.20261260

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Research article

Symmetric hybrid transformations for robust mean estimation under stratified systematic sampling

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Management Sciences, College of Business Administration, Hunan University, Changsha 410082, China
3.
Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
4.
Department of Statistics, University of Wah, Wah Cantt, Pakistan

Received: 08 May 2026 Revised: 30 August 2026 Accepted: 04 September 2026 Published: 29 September 2026
MSC : 62D05

Challenges arise in stratified random sampling when the auxiliary variable shows nonlinear behavior, pronounced skewness, or heterogeneous patterns across strata, leading to the reduced efficiency of traditional linear estimators. To overcome these limitations, this study develops a new class of weighted symmetric hybrid estimators for the finite population mean under a stratified design. The proposed methodology is constructed by combining harmonic, geometric, and fractional power transformations of auxiliary information within a unified symmetric framework, allowing the estimator to adapt to complex structural relationships while maintaining a balance between extreme and moderate observations within and across strata. Analytical expressions for bias and mean squared error are obtained, and optimal weights are derived through mean squared error minimization. Comparative theoretical analysis demonstrates that the proposed estimators consistently outperform classical ratio, product, regression, and exponential-type estimators. Furthermore, extensive simulation experiments conducted under various population structures, correlation levels, and stratification schemes demonstrate substantial efficiency enhancements, particularly for skewed and nonlinear populations. The proposed class therefore offers a flexible, symmetric, and robust alternative for practical implementation in stratified random sampling surveys.
Citation: Fatimah A. Almulhim, Umer Daraz, Hassan M. Aljohani, Javid Shabbir. Symmetric hybrid transformations for robust mean estimation under stratified systematic sampling[J]. AIMS Mathematics, 2026, 11(9): 32046-32080. doi: 10.3934/math.20261260

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Abstract

Challenges arise in stratified random sampling when the auxiliary variable shows nonlinear behavior, pronounced skewness, or heterogeneous patterns across strata, leading to the reduced efficiency of traditional linear estimators. To overcome these limitations, this study develops a new class of weighted symmetric hybrid estimators for the finite population mean under a stratified design. The proposed methodology is constructed by combining harmonic, geometric, and fractional power transformations of auxiliary information within a unified symmetric framework, allowing the estimator to adapt to complex structural relationships while maintaining a balance between extreme and moderate observations within and across strata. Analytical expressions for bias and mean squared error are obtained, and optimal weights are derived through mean squared error minimization. Comparative theoretical analysis demonstrates that the proposed estimators consistently outperform classical ratio, product, regression, and exponential-type estimators. Furthermore, extensive simulation experiments conducted under various population structures, correlation levels, and stratification schemes demonstrate substantial efficiency enhancements, particularly for skewed and nonlinear populations. The proposed class therefore offers a flexible, symmetric, and robust alternative for practical implementation in stratified random sampling surveys.

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