Optimal Debiased Inference on Privatized Data via Indirect Estimation and Parametric Bootstrap

We design a debiased parametric bootstrap framework for statistical inference from differentially private data. Existing usage of the parametric bootstrap on privatized data ignored or avoided handling the effect of clamping, a technique employed by the majority of privacy mechanisms. Ignoring the impact of clamping often leads to under-coverage of confidence intervals and miscalibrated type I errors of hypothesis tests. The main reason for the failure of the existing methods is the inconsistency of the parameter estimate based on the privatized data. We propose using the indirect inference method to estimate the parameter values consistently, and we use the improved estimator in parametric bootstrap for inference. To implement the indirect estimator, we present a novel simulation-based, adaptive approach along with the theory that establishes the consistency of the corresponding parametric bootstrap estimates, confidence intervals, and hypothesis tests. In particular, we prove that our adaptive indirect estimator achieves the minimum asymptotic variance among all "well-behaved" consistent estimators based on the released summary statistic. Our simulation studies show that our framework produces confidence intervals with well-calibrated coverage and performs hypothesis testing with the correct type I error, giving state-of-the-art performance for inference on location-scale normals, simple linear regression, and logistic regression.