Generalized family of estimators for imputing scrambled responses

When there is a high correlation between the study and auxiliary variables, the rank of the auxiliary variable also correlates with the study variable. Then, the use of the rank as an additional auxiliary variable may be helpful to increase the efficiency of the estimator of the mean or the total number of the population. In the present study, we propose two generalized families of estimators for imputing the scrambling responses by using the variance and the rank of the auxiliary variable. Expressions for the bias and the mean squared error are obtained up to the first order of approximation. A numerical study is carried out to observe the performance of estimators.