Variation from Classical Data Generating Procedures by Repeated Drawing

Consider \({y}_{i}\) takes value 1 if the \({i}^\text{th}(i=\text{1,2},\dots ,N)\) individual in the population \(U\) bears a sensitive characteristic \(A\) and value 0 if individual \(i\) bears \({A}^{C}\) . The proportion of individuals \(\theta =\frac{1}{N}\sum_{i=1}^{N}{y}_{i}\) in the population bearing \(A\) is estimated by employing randomized response (RR) devices pioneered by (Warner, Journal of American Statistical Association 60:63–69, 1965) followed by several other devices existing in literature. These devices usually mandate a sampled respondent to select card(s) randomly from a box answer a ‘match’ or ‘mismatch’ according to the characteristic ( \(A\) or \({A}^{C}\) ) marked in the selected card(s).