Estimation of the Mean of a Sensitive Variable in the Presence of Auxiliary Information

Abstract

Sousa et al. (2010) introduced a ratio estimator for the mean of a sensitive variable and showed that this estimator performs better than the ordinary mean estimator based on a randomized response technique (RRT). In this article, we introduce a regression estimator that performs better than the ratio estimator even for modest correlation between the primary and the auxiliary variables. The underlying assumption is that the primary variable is sensitive in nature but a non sensitive auxiliary variable exists that is positively correlated with the primary variable. Expressions for the Bias and MSE (Mean Square Error) are derived based on the first order of approximation. It is shown that the proposed regression estimator performs better than the ratio estimator and the ordinary RRT mean estimator (that does not utilize the auxiliary information). We also consider a generalized regression-cum-ratio estimator that has even smaller MSE. An extensive simulation study is presented to evaluate the performances of the proposed estimators in relation to other estimators in the study. The procedure is also applied to some financial data: purchase orders (a sensitive variable) and gross turnover (a non sensitive variable) in 2009 for a population of 5,336 companies in Portugal from a survey on Information and Communication Technologies (ICT) usage.

Keywords:

• Absolute relative bias
• Mean square error
• Randomized response technique
• Regression-cum-ratio estimator
• Regression estimator

Mathematics Subject Classification:

• 62D05

View correction statement:

Corrigendum to: Gupta et al. (2012). Estimation of the mean of a sensitive variable in the presence of auxiliary information. CSTM, 41 (13–14):2394–2404.

Acknowledgments

We would like to thank the two referees for making very constructive suggestions which resulted in a significant improvement over the original version of this article.

Notes

¹MSE comparison condition based on first-order approximation given in expression (4.10).

NACE is derived from the French title “Nomenclature générale des Activités économiques dans les Communautés Européennes” (Statistical classification of economic activities in the European Communities).

¹MSE comparison condition based on first-order approximation given in expression (4.10).