A two-step approach to ratio and regression estimation of finite population mean using optional randomized response models

We propose a modified two-step approach for estimating the mean of a sensitive variable using an additive optional RRT model which allows respondents the option of answering a quantitative sensitive question directly without using the additive scrambling if they find the question non-sensitive. This situation has been handled before in Gupta et al. (2010) using the split sample approach. In this work we avoid the split sample approach which requires larger total sample size. Instead, we estimate the finite population mean by using an Optional Additive Scrambling RRT Model but the corresponding sensitivity level is estimated from the same sample by using the traditional Binary Unrelated Question RRT Model of Greenberg et al. (1969). The initial mean estimation is further improved by utilizing information from a non-sensitive auxiliary variable by way of ratio and regression estimators. Expressions for the Bias and MSE of the proposed estimators (correct up to first order approximation) are derived. We compare the results of this new model with those of the split-sample based Optional Additive RRT Model of Kalucha et al. (2015), Gupta et al. (2015) and the simple optional additive RRT Model of Gupta et al. (2010). We see that the regression estimator for the new model has the smallest MSE among all of the estimators considered here when they have the same sample size.