Kernel Density Estimation for Joint Scrambling in Sensitive Surveys

Randomized response models aim to protect respondent privacy when sampling sensitive variables but consequently compromise estimator efficiency. We propose a new sampling method, titled joint scrambling, which preserves all true responses while protecting privacy by asking each respondent to jointly speak both their true response and multiple random responses in an arbitrary order. We give a kernel density estimator for the density function with asymptotically equivalent mean squared error for the optimal bandwidth yet greater generality than existing techniques for randomized response models. We also give consistent, unbiased estimators for a general class of estimands including the mean. For the cumulative distribution function, this estimator is more computationally efficient with asymptotically lower mean squared error than existing approaches. All results are verified via simulation and evaluated with respect to natural generalizations of existing privacy notions.