On the Analysis of Randomized Response Techniques under PPS Sampling Design

Muhammad Azeem

doi:10.64060/JASRv2i35

Authors

Muhammad Azeem Department of Statistics, University of Malakand, Khyber Pakhtunkhwa, Pakistan Author https://orcid.org/0000-0002-6475-6072

DOI:

https://doi.org/10.64060/JASRv2i35

Keywords:

Efficiency, Estimator, PPS Sampling, Privacy protection, Randomized Response

Abstract

In recent years, exploration of randomized response techniques has been a topic of interest among survey statisticians. Researchers use randomized response methods to obtain data related to sensitive characteristics e.g. exam cheating, consumption of drugs, illegal income, tax payment, and violation of laws, etc. The available randomized response methods use the conventional equal probability sampling methods where all units share equal probabilities of selection in the sample. In many real-life problems, the probability of selection may vary from unit to unit, making it impracticable to use equal probability sampling designs in such situations. In such cases, the PPS (probability proportional to size) sampling scheme becomes an appropriate technique for selection of samples. This study develops the mathematical framework for the analysis of optional randomized response technique (ORRT) models in PPS sampling. Additionally, an improved randomized response model using forced responses has also been presented under PPS sampling. The improvement in the suggested model over the competitor models has been observed under PPS sampling.

References

Ahmad, S., & Shabbir, J. (2018). Use of extreme values to estimate finite population mean under PPS sampling scheme. Journal of Reliability and Statistical Studies, 11(2), 99–112.

Ahmad, S., Zahid, E., Shabbir, J., Aamir, M., & Onyango, R. (2023a). Enhanced estimation of the population mean using two auxiliary variables under probability proportional to size sampling. Mathematical Problems in Engineering, Article ID 5564360. https://doi.org/10.1155/2023/5564360

Ahmad, S., Shabbir, J., Zahid, E., & Aamir, M. (2023b). Improved family of estimators for the population mean using supplementary variables under PPS sampling. Science Progress, 106(2): Article 00368504231180085. https://doi.org/10.1177/00368504231180085

Alomani, G., Kayid, M., & Ahmad, S. (2025). Application to radiation data sets by suggesting an improved mean estimator under probability proportional to size sampling. Journal of Radiation Research and Applied Sciences, 18, 101354. https://doi.org/10.1016/j.jrras.2025.101354

Azeem, M. (2023). Introducing a weighted measure of privacy and efficiency for comparison of quantitative randomized response models. Pakistan Journal of Statistics, 39(3), 377-385.

Azeem, M., & Ali, S. (2023). A neutral comparative analysis of additive, multiplicative, and mixed quantitative randomized response models. PLOS ONE, 18(4), e0284995. https://doi.org/10.1371/journal.pone.0284995

Bar-Lev, S. K., Bobovitch, E., & Boukai, B. (2004). A note on randomized response models for quantitative data. Metrika, 60(3), 255-260.

Diana, G., & Perri, P. F. (2011). A class of estimators of quantitative sensitive data. Statistical Papers, 52(3), 633-650. https://doi.org/10.1007/s00362-009-0273-1

Gjestvang, C. R., & Singh, S. (2009). An improved randomized response model: Estimation of mean. Journal of Applied Statistics, 36(12), 1361-1367. https://doi.org/10.1080/02664760802684151

Gupta, S., Mehta, S., Shabbir, J., & Khalil, S. (2018). A unified measure of respondent privacy and model efficiency in quantitative rrt models. Journal of Statistical Theory and Practice, 12(3), 506-511. https://doi.org/10.1080/15598608.2017.1415175

Gupta, S., Zhang, J., Khalil, S., & Sapra, P. (2024). Mitigating lack of trust in quantitative randomized response technique models. Communications in Statistics – Simulation and Computation, 53(6), 2624-2632. https://doi.org/10.1080/03610918.2022.2082477

Hansen, M. H., & Hurwitz, W. N. (1943). On the theory of sampling from a finite population. Annals of Mathematical Statistics, 14, 333-362.

Horvitz, D. G., & Thompson, D. J. (1952). A generalization of sampling without replacement from finite universe. Journal of the American Statistical Association, 47, 663-685.

Khan, S., Farooq, M., Ahmad, S., & Hussain, S. (2024). Improved estimator for estimation of population mean using predictive approach under PPS sampling. VFAST Transactions on Mathematics, 12(2), 1-16.

Narain, R. D. (1951). On sampling without replacement with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 3, 169-174.

Narjis, G., & Shabbir, J. (2023). An efficient new scrambled response model for estimating sensitive population mean in successive sampling. Communications in Statistics – Simulation and Computation, 52(11), 5327-5344. https://doi.org/10.1080/03610918.2021.1986528

Pushadapu, K., Singh, S., & Sedory, S. A. (2025). An optimized optional randomized response technique. International Statistical Review, 93(1), 130-149. https://doi.org/10.1111/insr.12581

Saleem, I., Sanaullah, A., Al-Essa, L. A., & Bashir, S. (2023). Efficient estimation of population variance of a sensitive variable using a new scrambling response model. Scientific Reports, 13, Article ID 19913. https://doi.org/10.1038/s41598-023-45427-2

Salemian, H., Mahmoudi, E., Alamri, O. A., & Shabbir, J. (2024). Improving a novel quantitative randomized response method using auxiliary variable information. Heliyon, 10, e40367. http://dx.doi.org/10.1016/j.heliyon.2024.e40367

Sen, A. R. (1953). On the estimate of the variance in sampling with varying probabilities. Journal of the Indian Society of Agricultural Statistics, 5, 119-127.

Sinha, R. R., & Khanna, B. (2002). Estimation of population mean under probability proportional to size sampling with and without measurement errors. Concurrency and Computation: Practice and Experience, 34(18), Article e7023.

Warner, S. L. (1971). The linear randomized response model. Journal of the American Statistical Association, 66(336), 884-888. https://doi.org/10.1080/01621459.1971.10482364

Yan, Z., Wang, J., & Lai, J. (2008). An efficiency and protection degree-based comparison among the quantitative randomized response strategies. Communications in Statistics – Theory and Methods, 38(3), 400-408.