estimating the prevalence of sensitive attribute with optional unrelated question randomized response models under simple and stratified random sampling

In this study, we propose optional randomized response technique (rrt) models in binary response situation. the utility of proposed optional rrt models under stratification are also explored. gupta et al.cite{singh} introduced an ingenious idea of optional rrt model, that a question may be sensitive for one respondent but may not be sensitive for another. this study focus on estimating $ pi $, the prevalence of sensitive attribute, $ omega $, the sensitivity level of the underlying sensitive question when the proportion of unrelated innocuous attribute $ pi_{{x}} $ is unknown. a new multi-question approach are proposed and used for estimation of parameters $ (pi,omega) $. a comparison between proposed optional rrt models and corresponding full rrt models are carried out numerically under simple and stratified random sampling.