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AIMS Mathematics

2026, Volume 11, Issue 6: 18692-18714. doi: 10.3934/math.2026760
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An improved randomized response model: Theory, efficiency analysis, and applications

  • 1.
    Department of Statistics, Quaid-i-Azam University, Islamabad 45320, Pakistan
  • 2.
    Department of Statistics, University of Sindh, Jamshoro, Pakistan
  • 3.
    Department of Statistics, University of Peshawar, Peshawar 25120, Pakistan
  • 4.
    Department of Mathematics and Statistics, College of Science, King Faisal University, Alahsa 31982, Saudi Arabia
  • 5.
    Department of Statistics, Federal University of Bahia, Salvador 40170-110, Brazil
  • 6.
    Department of Business Management, University of Pretoria, Pretoria 0002, South Africa
  • Published: 25 June 2026

  • MSC : 62D05, 62F10, 62P25

  • Randomized response (RR) techniques are commonly used to estimate the proportion of a sensitive attribute in a finite population while the preserving respondents' privacy. In this paper, we develop a novel two-stage randomized response model to jointly estimate the population proportion $ \eta $ of a sensitive characteristic and the probability $ T $ that a respondent provides a true answer under direct questioning. First the proposed design first implements direct questioning and only applies a randomization device to non-respondents, thus reducing unnecessary randomization and improving data quality. The randomization mechanism incorporates three additional questions within a two-stage probabilistic framework, thus offering enhanced privacy protection and improved statistical efficiency. Closed-form expressions for the proposed estimators are derived, and their statistical properties, including unbiasedness and variance, are obtained under simple random sampling. Analytical efficiency comparisons with existing models are established through mean-squared error criteria. The methodology is further extended to stratified random sampling, and the corresponding estimators and variance expressions are derived. Monte Carlo simulations are conducted to assess the finite-sample performance and to support the theoretical findings. The results demonstrate that the proposed estimators outperform competing approaches in both efficiency and privacy protection. The proposed framework provides a mathematically rigorous and practically effective tool for inference on sensitive characteristics in survey sampling.

    Citation: Shehzad Ahmad Khan, Moiz Qureshi, Hasnain Iftikhar, Fatimah E. Almuhayfith, Paulo Canas Rodrigues. An improved randomized response model: Theory, efficiency analysis, and applications[J]. AIMS Mathematics, 2026, 11(6): 18692-18714. doi: 10.3934/math.2026760

    Related Papers:

  • Randomized response (RR) techniques are commonly used to estimate the proportion of a sensitive attribute in a finite population while the preserving respondents' privacy. In this paper, we develop a novel two-stage randomized response model to jointly estimate the population proportion $ \eta $ of a sensitive characteristic and the probability $ T $ that a respondent provides a true answer under direct questioning. First the proposed design first implements direct questioning and only applies a randomization device to non-respondents, thus reducing unnecessary randomization and improving data quality. The randomization mechanism incorporates three additional questions within a two-stage probabilistic framework, thus offering enhanced privacy protection and improved statistical efficiency. Closed-form expressions for the proposed estimators are derived, and their statistical properties, including unbiasedness and variance, are obtained under simple random sampling. Analytical efficiency comparisons with existing models are established through mean-squared error criteria. The methodology is further extended to stratified random sampling, and the corresponding estimators and variance expressions are derived. Monte Carlo simulations are conducted to assess the finite-sample performance and to support the theoretical findings. The results demonstrate that the proposed estimators outperform competing approaches in both efficiency and privacy protection. The proposed framework provides a mathematically rigorous and practically effective tool for inference on sensitive characteristics in survey sampling.



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  • © 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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