Estimation of Mean of a Sensitive Quantitative Variable in Complex Survey: Improved Estimator and Scrambled Randomized Response Model

Abstract

With the intention to control a true swapping between the efficiency and the privacy protection this paper introduces a scrambled randomized response (SRR) model to be alternative of Saha’s scrambling mechanism. The basic initiative is to provide an assortment of the additive, the subtractive and the multiplicative models. The simulation and the empirical studies are provided for various sample sizes to compare the efficiency of the proposed model. The results obtained from simulation showed that the proposed model performs better than Pollock and Bek’s additive model. Also, the proposed generalized estimator of mean has been studied using a new SRR model presented in this article and shown that the proposed estimator and its class of estimators are more efficient than existing estimators. It is also shown that gain in efficiency is more when the proposed SRR model is used. The efficiency of the proposed class of estimators over existing estimators using both models is also provided using real data and with a simulation study.

Keywords

• Scrambled randomized response,sensitive variable,auxiliary variable,exponential estimator,mean square error,simulation

References

1. Asghar, A., Sanaullah, A, and Hanif, M. (2014). Generalized exponential type estimator for population variance in survey sampling. Revista Colombiana de Estadística, 37(1), 211-222.
2. Cheng C. C. and Singh, S. (2009). The Franklin's randomized response model for two sensitive attributes. Survey Research Methods, 4171-4184.
3. Diana, G., and Perri, P. F. (2010). New scrambled response models for estimating the mean of a sensitive quantitative character. Journal of Applied Statistics, 37(11), 1875-1890.
4. Diana, G., and Perri, P. F. (2011). A class of estimators for quantitative sensitive data. Statistical Papers, 52(3), 633-650.
5. Eichhorn, B.H. and Hayre, L. S. (1983). Scrambled randomized response models for obtaining sensitive quantitative data. Journal of Statistical Planning and Inference, 7, 307-316.
6. Greenberg, B. G., Kuebler, R. R., Jr., Abernathy, J. R., and Horvitz, D. G. (1971). Application of the randomized response technique in obtaining quantitative data. Journal of the American Statistical Association, 66, 243-250.
7. Gupta, S., Mehta, S., Shabbir, J., and Khalil, S. (2018). A unified measure of respondent privacy and model efficiency in quantitative RRT models. Journal of Statistical Theory and Practice, 1-6.
8. Gupta, S., Shabbir, J., Sousa, R., and Corte-real, P. (2012). Estimation of the mean of a sensitive variable in the presence of auxiliary information. Communications in Statistics-Theory and Methods, 41: 2394–2404.

9. Himmelfarb, S., and Edgell, S. E. (1980). Additive constants model: A randomized response technique for eliminating evasiveness to quantitative response questions. Psychological Bulletin, 87(3), 525.
10. Horvitz, D.G. and Thompson, D.J. (1952). A generalization of sampling without replacement from a finite universe. J. Amer. Statist. Assoc., 47, 663-685.
11. Hussain, Z. (2012). Improvement of the Gupta and Thornton scrambling model through double use of randomization device. International Journal Academic Research Business and Social Science, 2, 91-97.
12. Jabeen, R., Sanaullah, A., and Hanif, M. (2014). Generalized estimator for estimating population mean under two stage sampling. Pak. J. Statist., 30(4), 465-486.
13. Koyuncu, N., Gupta, S., and Sousa, R. (2014). Exponential-type estimators of the mean of a sensitive variable in the presence of non-sensitive auxiliary information. Communications in Statistics-Simulation and Computation, 43(7), 1583-1594.
14. Koyuncu, N., Kadilar, C. (2009). Family of estimators of population mean using two auxiliary variables in stratified random sampling. Communications in Statistics-Theory and Methods, 38(14), 2398–2417.
15. Pollock, K. H. and Bek, Y. (1976). A comparison of three randomized response models for quantitative data. Journal of the American Statistical Association, 71(356), 884-886.
16. Saha, A. (2008). A randomized response technique for quantitative data under unequal probability sampling. Journal of Statistical Theory and Practice, 2(4), 589-596.
17. Sanaullah, A., Ali, H.A., Noor ul Amin, M., and Hanif, M. (2014). Generalized exponential chain ratio estimators under stratified two-phase random sampling. Applied Mathematics and Computation, 226 541-547.
18. Sanaullah, A., Hanif, M and Asghar. A. (2016). Generalized exponential estimators for population variance under two-phase sampling. Int. J. Appl. Comput. Math., 2(1), 75-84.
19. Singh, H., and Tarray, T. (2014). An improved randomized response additive model. Sri Lankan Journal of Applied Statistics, 15(2), 131-138.
20. Sousa, R., Shabbir, J., Real, P. C., and Gupta, S. (2010). Ratio estimation of the mean of a sensitive variable in the presence of auxiliary information. Journal of Statistical Theory and Practice, 4(3), 495-507.
21. Tarray, T. A., and Singh, H. P. (2015). A general procedure for estimating the mean of a sensitive variable using auxiliary information. Investigacion Operacionel, 36(3), 249-262.
22. Warner, S. L. (1965). Randomized response: A survey technique for eliminating evasive answer bias. Journal of the American Statis tical Association, 60, 63-69.