Investigating the impact of simple and mixture priors on estimating sensitive proportion through a general class of randomized response models

Document Type : Article


1 Department of Statistics, Government College University, Faisalabad, 38000, Pakistan.; Department of Mathematics, Institute of Statistics, Zhejiang University, Hangzhou, 310027, China.

2 Government Degree College for Women, Samanabad, Faisalabad, 38000, Pakistan.

3 Department of Statistics, Quaid-i-Azam University, Islamabad, 44000, Pakistan.

4 Department of Mathematics and Statistics, King Fahad University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia.

5 Department of Statistics, Government College University, Faisalabad, 38000, Pakistan.


Randomized response is an efficacious and effective survey method to collect subtle information. It entitles respondents to respond to over-sensitive issues and defensive questions (such as criminal behavior, gambling habits, addiction to drugs, abortions, etc) while maintaining confidentiality. In this paper, we conducted a Bayesian analysis of a general class of randomized response models by using different prior distributions, such as Beta, Uniform, Jeffreys and Haldane, under squared error, precautionary and Degroot loss functions. We have also expanded our proposal for the case of mixture of Beta priors under squared error loss function. The performance of the Bayes and maximum likelihood estimators is evaluated in terms of mean squared errors. Moreover, an application with real data set is also provided to explain the proposal for practical considerations.


Main Subjects

1.Warner, S.L. Randomized response: A survey technique  for eliminating evasive answer bias", Journal  of the American Statistical Association, 60, pp. 63-69  (1965).
2. Greenberg, B., Abul-Ela, A., Simmons, W., and  Horvitz, D. The unrelated question randomized response:  theoretical framework", Journal of the American  Statistical Association, 64, pp. 529-539 (1969).
3. Moors, J.J.A. Optimizing of the unrelated question  randomized response model", Journal of the American  Statistical Association, 66, pp. 627-629 (1971).  4. Kim, M.J., Tebbs, J.M., and An, S.W. Extension  of Mangat's randomized response model", Journal of  Statistical Planning and Inference, 36(4), pp. 1554-  1567 (2006).  5. Christo_des, T.C. A generalized randomized response  technique", Metrika, 57, pp. 195-200 (2003).  6. Hussain, Z. and Shabbir, J. Improved estimation  procedures for the mean of sensitive variable using  randomized response model", Pakistan Journal of  Statistics, 25(2), pp. 205-220 (2009).  7. Kim, J.-M. and Heo, T.-Y. Randomized response  group testing model", Journal of Statistical Theory  and Practice, 7(1), pp. 33-48 (2013).  8. Lee, G.S., Hong, K-H., Kim, J-M., and Son, C-K. An  estimation of a sensitive attribute based on a two stage  strati_ed randomized response model with strati_ed  unequal probability sampling", Brazilian Journal of  Probability and Statistics, 28(3), pp. 381-408 (2014).  9. Abdelfatah, S. and Mazloum, R. Improved randomized  response model using three decks of cards", Model  Assisted Statistical Application, 9, pp. 63-72 (2014).  10. Tanveer, T.A. and Singh, H.P. Some improved additive  randomized response models utilizing higher  order moments ratios of scrambling variable", Models  Assisted Statistics and Applications, 10(4), pp. 361-  383 (2015).  11. Blair, G., Imai, K., and Zhou, Y-Y. Design and analysis  of the randomized response technique", Journal  of the American Statistical Association, 110(511), pp.  1304-1319 (2015).  12. Singh, H.P. and Gorey, S.M. An e_cient new randomized  response model", Communications in Statistics-  Theory and Methods, 46(24), pp. 12059-12074 (2017).  13. Chaudhuri, A. and Mukerjee, R., Randomized Response:  Theory and Methods, Marcel- Decker, New  York (1998).  14. Tracy, D. and Mangat, N. Some development in  randomized response sampling during the last decadea  follow up of review by Chaudhuri and Mukerjee",  Journal of Applied Statistical Science, 4, pp. 533-544  (1996).  15. Chaudhuri, A., Randomized Response and Indirect  Questioning Techniques in Surveys, Chapman & Hall,  CRC Press, Taylor & Francis, Boca Raton, USA  (2011).  16. Chaudhuri, A. and Christo_des, T.C. Indirect Questioning  in Sample Surveys, Springer Verlag, Berlin  Heidelberg (2013).  17. Winkler, R. and Franklin, L. Warner's randomized  response model: A Bayesian approach", Journal of  the American Statistical Association, 74, pp. 207-214  (1979).  18. Migon, H. and Tachibana, V. Bayesian approximations  in randomized response models", Computational  Statistics and Data Analysis, 24, pp. 401-409 (1997).  19. Pitz, G. Bayesian analysis of randomized response  models", Journal of Psychological Bulletin, 87, pp.  209-212 (1980).  20. O'Hagan, A. Bayes linear estimators for randomized  response models", Journal of the American Statistical  Association, 82, pp. 580-585 (1987).  21. Spurrier, J. and Padgett, W. The application of  Bayesian techniques in randomized response", Sociological  Methods, 11, pp. 533-544 (1980).  22. Oh, M. Bayesian analysis of randomized response  models: a Gibbs sampling approach", Journal of the  Korean Statistical Society, 23, pp. 463-482 (1994).  23. Unnikrishnan, N. and Kunte, S. Bayesian analysis for  randomized response models", Sankhya, 61, pp. 422-  432 (1999).  24. Barabesi, L. and Marcheselli, M. A practical implementation  and Bayesian estimation in Franklin's  randomized response procedure", Communications is  Statistics-Simulations and Computation, 35, pp. 365-  373 (2006).  25. Barabesi, L. and Marcheselli, M. Bayesian estimation  of proportion and sensitivity level in randomized response  procedures", Metrika, 72, pp. 75-88 (2010).  26. Hussain, Z. and Shabbir, J. Bayesian estimation of  population proportion of a sensitive characteristic using  simple Beta prior", Pakistan Journal of Statistics,  25(1), pp. 27-35 (2009).  1022 M. Abid et al./Scientia Iranica, Transactions E: Industrial Engineering 26 (2019) 1009{1022  27. Hussain, Z. and Shabbir, J. Bayesian estimation of  population proportion in Kim andWarde (2005) mixed  randomized response using mixed prior distribution",  Journal of Probability and Statistical Science, 7(1), pp.  71-80 (2009).  28. Hussain, Z. and Shabbir, J. Estimation of the mean  of a socially undesirable characteristics", Scientia Iranica,  20(3), pp. 839-845 (2013).  29. Hussain, Z., Shabbir, J., and Riaz, M. Bayesian  estimation using Warner's randomized response model  through simple and mixture prior distributions", Communications  in Statistics-Simulations and Computations,  40(1), pp. 147-164 (2011).  30. Bar-Lev, E.S., Bobovich, K., and Boukai, B. A common  conjugate prior structure for several randomized  response models", Test, 12(1), pp. 101-113 (2003).  31. Adepetun, A.O. and Adewara, A.A. Bayesian analysis  of Kim and Warde randomized response technique  using alternative priors", American Journal of Computational  and Applied Mathematics, 4(4), pp. 130-140  (2014).  32. Hussain, Z., Abid, M., Shabbir, J., and Abbas, N.  On Bayesian analysis of a general class of randomized  response models in social surveys about stigmatized  traits", Journal of Data Science, 12(4), pp. 1-21  (2014).  33. Son, C.-K. and Kim, J.-M. Bayes linear estimators of  two stage and strati_ed randomized response models",  Models Assisted Statistics and Applications, 10, pp.  321-333 (2015).  34. Song, J.J. and Kim, J-M. Bayesian estimation of rare  sensitive attribute", Communications in Statistics-  Simulation and Computation, 46(5), pp. 4154-4160  (2017).  35. Legendre, A., New Methods for the Determination of  Orbits of Comets, Courcier, Paris (1805).  36. Gauss, C.F., Least Squares Method for the Combinations  of Observations, (translated by J. Bertrand  1955), Mallet-Bachelier, Paris (1810).  37. Norstrom, J.G. The use of precautionary loss functions  in risk analysis", IEEE Transaction Reliability,  45, pp. 400-403 (1996).  38. DeGroot, M.H., Optimal Statistical Decision,  McGraw-Hill Inc, New York (1970).  39. Chaubey, Y. and Li, W. Comparison between maximum  likelihood and Bayes methods of estimation  for binomial probability with sample compositing",  Journal of O_cial Statistics, 11, pp. 379-390 (1995).  40. Liu, P.T. and Chow, L.P. The e_ciency of the multiple  trial randomized response model", Biometrics, 32,  pp. 607-618 (1976).