, Volume 46, Issue 1, pp 5–22 | Cite as

A generalized procedure for estimating the multinomial proportions in randomized response sampling using scrambling variables

  • Housila P. Singh
  • Swarangi M. GoreyEmail author
Original Paper


Taking clue from pioneer work of Chen and Singh (J Mod Appl Stat Methods 11:105–122, 2012), we have suggested a generalized RR procedure, for estimating the multinomial proportions of potentially sensitive attributes in survey sampling, using higher order moments of scrambling variables at the estimation stage to produce unbiased estimators. The RR procedure due to Chen and Singh (J Mod Appl Stat Methods 11:105–122, 2012) is viewed as member of the proposed RR procedure. Expressions for variance and covariance of the suggested generalized estimator with its development are derived. It is found that the developed estimator is more efficient than Warner’s (J Am Stat Assoc 60:63–69, 1965) RRT. A numerical illustration is also given in support of the present study.


Multinomial proportions Unbiased estimators Randomized response sampling Scrambling variables Efficiency 

AMS Subject Classification




Authors are thankful to the Editor and the learned referee for the kind perusal of the paper and the suggestions given regarding improvement of the paper.


  1. Abernathy JR, Greenberg BG, Horvitz DG (1970) Estimates of induced abortion in urban North Carolina. Demography 7:19–29CrossRefGoogle Scholar
  2. Abul-Ela ALA, Greenberg BG, Horvitz DG (1967) A multi-proportions randomized response model. J Am Stat Assoc 62:990–1008MathSciNetCrossRefGoogle Scholar
  3. Antonak RF, Livneh H (1995) Direct and indirect methods to measure attitudes toward persons with disabilities, with an exegesis of the error-choice test method. Rehabil Psychol 40(1):3–24CrossRefGoogle Scholar
  4. Bourke PD (1981) On the analysis of some multivariate randomized response design of categorical data. J Stat Plan Inference 5:165–170MathSciNetCrossRefzbMATHGoogle Scholar
  5. Bourke PD (1982) RR multivariate designs for categorical data. Commun Stat Theory Methods 11:2889–2901CrossRefzbMATHGoogle Scholar
  6. Bourke PD, Dalenius T (1976) Some new ideas in the realm of randomized inquiries. Int Stat Rev 44:219–221CrossRefzbMATHGoogle Scholar
  7. Bradburn N, Sudman S (1979) Improving interview method and questionnaire design. Jossey-Bass, San-Francisco, pp 1–13Google Scholar
  8. Chaudhuri A, Mukerjee R (1988) Randomized response: theory and techniques. Marcel-Dekker, New YorkzbMATHGoogle Scholar
  9. Chen CC, Singh S (2012) Estimation of multinomial proportions using higher order moments of scrambling variables in randomized response sampling. J Mod Appl Stat Methods 11(1):105–122CrossRefGoogle Scholar
  10. Denmark B, Swensson B (1987) Measuring drug use among Swedish adolescents. J Off Stat 3:439–448Google Scholar
  11. Duffy JC, Waterton JJ (1984) Randomized response models for estimating the distribution function of a quantitative character. Int Stat Rev 52:165–172CrossRefGoogle Scholar
  12. Eichhorn BH, Hayre LS (1983) Scrambled randomized response methods for obtaining sensitive quantitative data. J Stat Plan Inference 7:307–316CrossRefGoogle Scholar
  13. Eriksson SA (1973) A new model for randomized response. Int Stat Rev 41:101–113CrossRefzbMATHGoogle Scholar
  14. Fox JA, Tracy PE (1986) Randomized response: a method of sensitive surveys. SEGE Publications, Newbury ParkCrossRefGoogle Scholar
  15. Himmelfarb S, Edgell SE (1980) Additive constants model: a randomized response technique for eliminating evasiveness to quantitative response questions. Physiol Bull 87:525–530Google Scholar
  16. Kerkvliet J (1994) Cheating by economics students: a comparison of survey results. J Econ Educ 25:121–133CrossRefGoogle Scholar
  17. Kim JJ, Flueck JA (1978) An additive randomized response model. In: Proceedings of the survey research methods section. American Statistical association, pp 351–355Google Scholar
  18. Kim JM, Warde DW (2005) A mixed randomized response model. J Stat Plan Inference 133(1):211–221MathSciNetCrossRefzbMATHGoogle Scholar
  19. Liu PT, Chow LP (1976) A new discrete quantitative randomized response model. J Am Stat Assoc 71:72–73CrossRefzbMATHGoogle Scholar
  20. Pollock KH, Bek Y (1976) A comparison of three randomized response models for quantitative data. J Am Stat Assoc 71:884–886CrossRefzbMATHGoogle Scholar
  21. Scheers NJ (1992) A review of randomized response technique. Meas Eval Couns Dev 25:27–41Google Scholar
  22. Tracy PE, Fox JA (1981) The validity of randomized response for sensitive measurement. Am Sociol Rev 46:187–200CrossRefGoogle Scholar
  23. Umesh UA, Peterson RA (1991) A critical evolution of randomized response method. Sociol Method Res 20(1):104–138CrossRefGoogle Scholar
  24. Warner SL (1965) Randomized response: a survey technique for eliminating evasive answer bias. J Am Stat Assoc 60:63–69CrossRefzbMATHGoogle Scholar

Copyright information

© The Behaviormetric Society 2018

Authors and Affiliations

  1. 1.School of Studies in StatisticsVikram UniversityUjjainIndia

Personalised recommendations