@article{Khan2026, 
author = {Shehzad Ahmad Khan and Moiz Qureshi and Hasnain Iftikhar and Fatimah E. Almuhayfith and Paulo Canas Rodrigues},
title = {An improved randomized response model: Theory, efficiency analysis, and applications},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {18692-18714},
keywords = {randomized response technique, sensitive survey, two-stage randomization, stratified sampling, unbiased estimation, efficiency comparison},
url = {https://www.sciopen.com/article/10.3934/math.2026760},
doi = {10.3934/math.2026760},
abstract = {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    η 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.}
}