  In some socio-economic surveys, data are collected on sensitive or
stigmatizing issues such as tax evasion, criminal conviction, drug use, etc. In
such surveys, direct questioning of respondents is not of much use and the
randomized response technique is used instead. A few researchers have studied
the issue of privacy protection or respondent jeopardy for surveys on
dichotomous populations, where the objective is to estimate the proportion of
persons bearing the sensitive trait. However, not much is yet known about
respondent protection when the variable under study takes discrete numerical
values and the objective of the survey is to estimate the population mean of
this variable. In this article we study this issue. We first propose a
randomization device for this situation and give the corresponding estimation
procedure. We next propose a measure of privacy and show that given a certain
stipulated level of this privacy measure, we can determine the parameter of the
randomization device so as to maximize the efficiency of estimation, while
guaranteeing the desired level of privacy protection. In particular, our study
also covers the case of polychotomous populations and we can estimate the
proportions of individuals belonging to the different classes. Consequently,
results for dichotomous populations follow as corollaries.
