Design-based distribution function estimation for stigmatized populations

In this paper, we discuss in a general framework the design-based estimation of population parameters when sensitive data are collected by randomized response techniques. We show in close detail the procedure for estimating the distribution function of a sensitive quantitative variable and how to estimate simultaneously the population prevalence of individuals bearing a stigmatizing attribute and the distribution function for the members belonging to the hidden group. The randomized response devices by Greenberg et al. (J Am Stat Assoc 66:243–250, 1971), Franklin (Commun Stat Theory Methods 18:489–505, 1989), and Singh et al. (Aust NZ J Stat 40:291–297 1998) are here considered as data-gathering tools.