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

Document Type : Article

Authors

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.

Abstract

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.

Keywords

Main Subjects

References

References:
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. Christofides, 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 stratified randomized response model with stratified 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 efficient 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 Christofides, 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).
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 stratified 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 Official Statistics, 11, pp. 379-390 (1995).
40. Liu, P.T. and Chow, L.P. "The efficiency of the multiple trial randomized response model", Biometrics, 32, pp. 607-618 (1976).
Scientia Iranica
Volume 26, Issue 2 - Serial Number 2
Transactions on Industrial Engineering (E)
March and April 2019
Pages 1009-1022

Files

Share

How to cite

Statistics