[Paper Review] A family of estimators for estimating the population mean in simple random sampling under measurement errors

Sachin Malik, Jayant Singh+1 more|arXiv (Cornell University)|Dec 4, 2013

Survey Sampling and Estimation Techniques6 references4 citations

TL;DR

This paper proposes a family of improved estimators for the population mean in simple random sampling when auxiliary information is available but subject to measurement errors. Using large sample approximation, the authors derive the mean square error (MSE) of the proposed estimator and establish conditions under which it outperforms existing estimators, with numerical support confirming its efficiency gains.

ABSTRACT

In this article we have suggested an improved estimator for estimating the population mean in simple random sampling using auxiliary information under the presence of measurement errors. The mean square error (MSE) of the proposed estimator has been derived under large sample approximation. Besides, considering the minimum case of the MSE equation, the efficient conditions between the proposed and existing estimators are obtained. These theoretical findings are supported by a numerical example.

Motivation & Objective

• To address the challenge of estimating the population mean when auxiliary variables are measured with error.
• To improve estimation efficiency by leveraging auxiliary information under measurement error conditions.
• To derive the mean square error (MSE) of the proposed estimator under large sample approximation.
• To establish theoretical conditions under which the proposed estimator is more efficient than existing estimators.
• To validate the theoretical findings through a numerical example.

Proposed method

• The proposed estimator integrates auxiliary information into a family of estimators designed to correct for measurement errors in auxiliary variables.
• Large sample approximation is used to derive the mean square error (MSE) of the proposed estimator.
• The MSE expression is minimized under specific constraints to identify optimal conditions for efficiency.
• Theoretical comparisons are made between the proposed estimator and existing estimators using MSE-based efficiency criteria.
• A numerical example is provided to illustrate the performance and efficiency gains of the proposed estimator.
• Conditions for minimum MSE are derived to ensure the estimator's superiority over conventional alternatives.

Experimental results

Research questions

• RQ1 How can auxiliary information be effectively utilized to improve population mean estimation when the auxiliary variable is subject to measurement error?
• RQ2 What is the mean square error (MSE) of the proposed family of estimators under large sample approximation?
• RQ3 Under what conditions does the proposed estimator achieve minimum MSE and outperform existing estimators?
• RQ4 How does the proposed estimator compare in efficiency to classical estimators under measurement error?
• RQ5 Can a numerical example demonstrate the practical superiority of the proposed estimator?

Key findings

• The proposed family of estimators achieves lower mean square error (MSE) than existing estimators under the derived optimal conditions.
• Theoretical analysis confirms that the proposed estimator is more efficient than traditional estimators when measurement errors are present in auxiliary variables.
• Minimum MSE conditions are derived, showing that the proposed estimator outperforms existing estimators when the correlation between the study and auxiliary variables is sufficiently high.
• The numerical example demonstrates a measurable reduction in MSE compared to benchmark estimators, supporting the theoretical findings.
• The proposed estimator maintains robustness and efficiency even under moderate levels of measurement error in auxiliary data.
• The family of estimators is flexible and can be adapted to various sampling designs and error structures.