Data Science Perspectives on Calibration Estimators for Finite Population Mean in Stratified Random Sampling Using Different Distance Measures

Abstract

This article presents a data analytics framework that utilizes auxiliary information to improve estimation accuracy and decision-making in stratified sampling. Various analytical models based on different distance measures are developed and evaluated to optimize data-driven insights. The study discusses the theoretical foundations of the proposed estimation techniques and derives optimized weighting schemes under different distance measures. The performance of the developed methods is assessed through simulation studies and comparative analysis. Furthermore, a real-world dataset is analyzed to demonstrate the effectiveness, robustness, and practical applicability of the proposed data analytics approach. The results indicate that the suggested methodology provides superior accuracy and efficiency compared to conventional techniques, making it a valuable tool for modern data-driven applications.

Introduction

This paper focuses on improving the precision of population mean estimation in stratified random sampling by developing calibration estimators under different distance measures using auxiliary information. Stratified sampling is one of the most widely used survey sampling techniques because it increases the accuracy of estimates by dividing the population into homogeneous strata. Calibration estimation further enhances survey efficiency by adjusting the original design weights using known auxiliary information while maintaining consistency with population totals.

The calibration approach works by replacing the initial sampling weights with calibrated weights, which are obtained by minimizing a selected distance function subject to calibration constraints. Since the pioneering work of Deville and Särndal (1992), many researchers have proposed different calibration estimators and distance measures to improve estimation under various sampling designs. Studies by Estevao and Särndal, Arnab and Singh, Farrell and Singh, Kim and Park, Tracy et al., Koyuncu and Kadilar, Rai et al., and others have demonstrated that calibration methods can substantially improve estimator efficiency in stratified random sampling.

The primary objective of this research is to develop new calibration estimators under stratified random sampling by employing different distance measures together with auxiliary information. The study first establishes the mathematical framework, introduces the necessary notations, formulates calibration estimators using alternative distance functions, and finally evaluates their performance through a simulation study based on real census data from Agra District, Uttar Pradesh (Census Series 10, Part 12B).

The paper considers a finite population divided into L strata, where both the study variable and an auxiliary variable are observed. Standard statistical quantities such as stratum means, variances, covariances, correlation coefficients, and stratum weights are defined to formulate the calibration problem. The traditional calibration estimator proposed by Tracy et al. (2003) is adopted as the starting point, with calibrated weights replacing the original stratum weights.

The literature review summarizes earlier work on calibration estimation using various distance measures. Koyuncu and Kadilar (2013) proposed four alternative distance functions and found that calibration based on the first distance measure performed better than the third. Rai et al. (2021) extended this work by incorporating two auxiliary variables and introducing five different distance measures. Building upon these studies, the present paper proposes four modified distance measures denoted as L1pL_{1p}L1p?, L2pL_{2p}L2p?, L3pL_{3p}L3p?, and L4pL_{4p}L4p?. These distance functions combine features of previously established measures while incorporating additional calibration constraints.

Unlike earlier approaches that mainly calibrated weights using the auxiliary variable mean, this study introduces two calibration constraints simultaneously: one based on the known auxiliary variable mean and another based on the correlation coefficient between the study and auxiliary variables. These additional constraints are expected to improve the efficiency of the resulting estimator by utilizing more auxiliary information.

The estimation problem is formulated as a constrained optimization problem. Using the Lagrange multiplier method, the calibrated weights are obtained by minimizing each proposed distance function while satisfying the calibration equations. The derivation begins with the first proposed distance measure, where the optimal calibrated weights are expressed as functions of the original design weights, auxiliary variable means, and correlation coefficients. Simultaneous equations are solved to estimate the Lagrange multipliers, which are then substituted into the calibration weight expressions. This process leads to the development of a new calibration estimator that adjusts the conventional stratified estimator using correction terms based on differences between sample and population auxiliary information.

The paper also begins the derivation of a second estimator based on another distance measure, following a similar optimization procedure. In this case, the calibrated weights differ from the first estimator because the selected distance function modifies the penalty associated with deviations from the original design weights. Although only the initial derivation is presented, the same calibration framework is intended to be applied to all proposed distance measures.

Conclusion

In this study we worked on new weights using different distance measures in stratified sampling. The performance of distance measures are compared with a simulation study.

References

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Copyright

Copyright © 2026 Anupama Bhadauriya, Rakesh Prasad Sarang, Anupam Kumar Srivastava. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.