Estimating a sensitive proportion through randomized response procedures based on auxiliary information

Randomized response techniques are widely employed in surveys dealing with sensitive questions to ensure interviewee anonymity and reduce nonrespondents rates and biased responses. Since Warner\u2019s (J Am Stat Assoc 60:63\u201369, 1965) pioneering work, many ingenious devices have been suggested to increase respondent\u2019s privacy protection and to better estimate the proportion of people, \u3c0A, bearing a sensitive attribute. In spite of the massive use of auxiliary information in the estimation of non-sensitive parameters, very few attempts have been made to improve randomization strategy performance when auxiliary variables are available. Moving from Zaizai\u2019s (Model Assist Stat Appl 1:125\u2013130, 2006) recent work, in this paper we provide a class of estimators for \u3c0A, for a generic randomization scheme, when the mean of a supplementary non-sensitive variable is known. The minimum attainable variance bound of the class is obtained and the best estimator is also identified. We prove that the best estimator acts as a regression-type estimator which is at least as efficient as the corresponding estimator evaluated without allowing for the auxiliary variable. The general results are then applied to Warner and Simmons\u2019 model