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METRON

, Volume 76, Issue 2, pp 203–230 | Cite as

Dealing sensitive characters on successive occasions through a general class of estimators using scrambled response techniques

  • Kumari PriyankaEmail author
  • Pidugu Trisandhya
  • Richa Mittal
Article

Abstract

Present article endeavours to propose a general class of estimators to estimate population mean of a sensitive character using non-sensitive auxiliary information under five different scrambled response models in two occasions successive sampling. Various well-known estimators have been modified for the estimation of sensitive population mean and hence they also become a member of proposed general class of estimators. Properties of proposed class of estimators have been derived and checked empirically while comparing the proposed class of estimators with respect to modified Jessen (Iowa Agric Exp Stn Res Bull 304:1–104, 1942) type estimator and modified Singh (Stat Transit 7(1):21–26, 2005) type estimator under five different scrambled response models. The effectiveness of different models has been discussed while comparing it with the direct questioning methods. A model for optimum total cost has also been proposed. Privacy protection has been elaborated for all considered models. Numerical illustrations including simulation studies are abundant to the theoretical results. Finally suitable recommendations are forwarded.

Keywords

Successive sampling Scrambled response model Sensitive character Class of estimators Population mean Bias Mean squared error Optimum replacement policy 

Mathematics Subject Classification

62D05 

Notes

Acknowledgements

The authors wish to thank anonymous reviewer and Professor Giovanni Maria Giorgi, Editor in Chief for their careful reading and constructive suggestions which lead to improvement over an earlier version of the paper.

References

  1. 1.
    Arnab, R.: Alternative estimators for randomized response techniques in multi-character surveys. Commun. Stat. Theory Methods 40(10), 1839–1848 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arnab, R., Singh, S., North, D.: Use of two decks of cards in randomized response techniques for complex survey designs. Commun. Stat. Theory Methods 41(16–17), 3198–3210 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arnab, R., Singh, S.: Estimation of mean of sensitive characteristics for successive sampling. Commun. Stat. Theory Methods 42(14), 2499–2524 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bandhyopadhyay, A., Singh, G.N.: On the use of two auxiliary variables to improve the precision of estimate in two-occasion successive sampling. Int. J. Math. Stat. 15(1), 73–88 (2014)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Christofides, T.C.: A generalized randomized response technique. Metrika 57(2), 195–200 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Christofides, T.C.: Randomized response in stratified sampling. J. Stat. Plan. Inference 128(1), 303–310 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cochran, W.C.: Sampling Techniques, 3rd edn. Wiley, New York (1977)zbMATHGoogle Scholar
  8. 8.
    Diana, G., Perri, P.F.: New scrambled response models for estimating the mean of a sensitive quantitative character. J. Appl. Stat. 37(11), 1875–1890 (2010)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Diana, G., Perri, P.F.: A class of estimators for quantitative sensitive data. Stat. Pap. 52(3), 633–650 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Eichhorn, B.H., Hayre, L.S.: Scrambled randomized response method for obtaining sensitive quantitative data. J. Stat. Plan. Inference 7(4), 307–316 (1983)CrossRefGoogle Scholar
  11. 11.
    Feng, S., Zou, G.: Sample rotation method with auxiliary variable. Commun. Stat. Theory Methods 26(6), 1497–1509 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Greenberg, B.G., Kubler, R.R., Horvitz, D.G.: Application of RR technique in obtaining quantitative data. J. Am. Stat. Assoc. 66(334), 243–250 (1971)CrossRefGoogle Scholar
  13. 13.
    Gupta, S., Gupta, B., Singh, S.: Estimation of sensitivity level of personal interview survey question. J. Stat. Plan. Inference 100(2), 239–247 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gupta, S., Shabbir, J.: Sensitivity estimation for personal interview survey questions. Statistica 64(4), 643–653 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Gupta, S.N., Thorton, B., Shabbir, J., Singhal, S.: A comparison of multiplicative and additive optional RRT models. J. Stat. Theory Appl. 5(3), 226–239 (2006)MathSciNetGoogle Scholar
  16. 16.
    Gupta, S., Shabbir, J.: On improvement in estimating the population mean in simple random sampling. J. Appl. Stat. 35(5), 559–566 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gupta, S., Shabbir, J., Sousa, R., Real, P.C.: Estimation of the mean of a sensitive variable in the presence of auxiliary information. Commun. Stat. Theory Methods 41(13–14), 2394–2404 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Horvitz, D.G., Shah, B.V., Simmons, W.R.: The unrelated question randomized response model. In: Proc. Soci. Statist. Amer. Statist. Asso., pp. 65–72 (1967)Google Scholar
  19. 19.
    Hussain, Z., Al-Zahrani, B.: Mean and sensitivity estimation of a sensitive variable through additive scrambling. Commun. Stat. Theory Methods 45(1), 182–193 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Jessen, R.J.: Statistical investigation of a sample survey for obtaining farm facts. Iowa Agric. Exp. Stn. Res. Bull. 304, 1–104 (1942)Google Scholar
  21. 21.
    Kim, J.-M., Elam, M.E.: A stratified unrelated question randomized response model. Stat. Pap. 48(2), 215–233 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Koyuncu, N., Gupta, S., Sousa, R.: Exponential-type estimators of the mean of a sensitive variable in the presence of a non-sensitive auxiliary information. Commun. Stat. Simul. Comput. 43(7), 1583–1594 (2014)CrossRefzbMATHGoogle Scholar
  23. 23.
    Murthy, M.N.: Sampling Theory and Methods. Statistical Publication Society, Calcutta (1967)zbMATHGoogle Scholar
  24. 24.
    Naeem, N., Shabbir, J.: Use of scrambled responses on two occasions successive sampling under non-response. Hacettepe University Bulletin of Natural Sciences and Engineering Series B: Mathematics and Statistics, p. 46 (2016). http://www.hjms.hacettepe.edu.tr/uploads/32c18b80-275f-4b5a-a28d-6a5890eecac3.pdf
  25. 25.
    Patterson, H.D.: Sampling on successive occasions with partial replacement of units. J. R. Stat. Soc. Ser. B 12, 241–255 (1950)zbMATHGoogle Scholar
  26. 26.
    Pollock, K.H., Bek, Y.: A comparison of three randomized response models for quantitative data. J. Am. Stat. Assoc. 71(336), 884–886 (1976)CrossRefzbMATHGoogle Scholar
  27. 27.
    Perri, P.F., Diana, G.: Scrambled response models Based on auxiliary variables. In: Torelli, N., Pesarin, F., Bar-Hen, A. (eds.) Advances in Theoretical and Applied Statistics, pp. 281–291. Springer, Berlin (2013)CrossRefGoogle Scholar
  28. 28.
    Priyanka, K., Mittal, R.: Effective rotation patterns for median estimation in successive sampling. Stat. Trans. 15(2), 197–220 (2014)Google Scholar
  29. 29.
    Priyanka, K., Mittal, R., Min-Kim, J.: Multivariate rotation design for population mean in sampling on successive occasions. Commun. Stat. Appl. Methods 22(5), 445–462 (2015)Google Scholar
  30. 30.
    Priyanka, K., Mittal, R.: A class of estimators for population median in two occasion rotation sampling. HJMS 44(1), 189–202 (2015a)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Priyanka, K., Mittal, R.: Estimation of population median in two-occasion rotation sampling. J. Stat. Appl. Probab. Lett. 2(3), 205–219 (2015b)zbMATHGoogle Scholar
  32. 32.
    Reddy, V.N.: A study on the use of prior knowledge on certain population parameters in estimation. Sankhya C 40, 29–37 (1978)zbMATHGoogle Scholar
  33. 33.
    Srivastava, S.K.: A generalized estimator for the mean of a finite population using multi auxiliary information. J. Am. Stat. Assoc. 66, 404–407 (1971)CrossRefzbMATHGoogle Scholar
  34. 34.
    Sen, A.R.: Theory and application of sampling on repeated occasions with several auxiliary variables. Biometrics 29(2), 381–385 (1973)CrossRefGoogle Scholar
  35. 35.
    Srivastava, S.K., Jhajj, H.S.: A class of estimators using auxiliary information for estimating finite population variance. Sankhya Ser. C 42, 87–96 (1980)zbMATHGoogle Scholar
  36. 36.
    Sukhatme, P.V., Sukhatme B.V., Sukhatme S., Ashok, C.: Sampling theory of surveys with applications. The Indian society of Agricultural Statistics, New Delhi. The lowa state college press, Ames, pp. xxix+491 (1984)Google Scholar
  37. 37.
    Singh, H.P., Ruiz-Espejo, M.R.: On linear regression and ratio-product estimation of finite population mean. Statistician 52(1), 59–67 (2003)MathSciNetGoogle Scholar
  38. 38.
    Singh, G.N.: On the use of chain-type ratio estimator in successive sampling. Stat. Transit. 7(1), 21–26 (2005)MathSciNetGoogle Scholar
  39. 39.
    Saha, A.: A simple randomized response technique in complex surveys. Metron LXV(1), 59–66 (2007)MathSciNetGoogle Scholar
  40. 40.
    Singh, H.P., Vishwakarma, G.K.: A general class of estimators in successive sampling. Metron LXV(2), 201–227 (2007)MathSciNetGoogle Scholar
  41. 41.
    Singh, G.N., Priyanka, K.: Search of good rotation patterns to improve the precision of estimates at current occasion. Commun. Stat. Theory Methods 37(3–5), 337–348 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Singh, S., Sedory, S.A.: A true simulation study of three estimators at equal protection of respondents in randomized response sampling. Stat. Neer. 66(4), 442–451 (2012)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Sihm, J.-S., Gupta, S.: A two-stage binary optional randomized response model. Commun. Stat. Simul. Comput. 44(9), 2278–2296 (2015)MathSciNetCrossRefGoogle Scholar
  44. 44.
    Tracy, D.S., Singh, H.P., Singh, R.: An alternative to the ratio-cum-product estimator in sample surveys. J. Stat. Plan. Inference 53, 375–387 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Warner, S.L.: Randomized response: a survey technique for eliminating evasive answer bias. J. Am. Stat. Assoc. 60(309), 63–69 (1965)CrossRefzbMATHGoogle Scholar
  46. 46.
    Warner, S.L.: The linear randomized response model. J. Am. Stat. Assoc. 66(336), 884–888 (1971)CrossRefGoogle Scholar
  47. 47.
    Wu, J.-W., Tian, G.-L., Tang, M.-L.: Two new models for survey sampling with sensitive characteristics: design and analysis. Metrika 67, 251–263 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Yan, Z., Wang, J., Lai, J.: An efficiency and protection degree-based comparison among the quantitative randomized response strategies. Commun. Stat. Theory Methods 38(3–5), 400–408 (2009)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Yu, B., Jin, Z., Tian, J., Gao, G.: Estimation of sensitive proportion by randomized response data in successive sampling. Comput. Math. Methods Med. (2015).  https://doi.org/10.1155/2015/172918

Copyright information

© Sapienza Università di Roma 2017

Authors and Affiliations

  • Kumari Priyanka
    • 1
    Email author
  • Pidugu Trisandhya
    • 1
  • Richa Mittal
    • 2
  1. 1.Department of Mathematics, Shivaji CollegeUniversity of DelhiNew DelhiIndia
  2. 2.Department of MathematicsNIT CalicutCalicutIndia

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