Empirical Likelihood for the Differentially Private Mean

We develop a nonparametric framework for confidence intervals for a population mean under differential privacy (DP), without requiring variance estimation or a parametric model for the underlying distribution. We partition clipped observations into disjoint blocks, privatize each block mean with Laplace noise, and construct confidence intervals using empirical likelihood (EL) and adjusted empirical likelihood (AEL). We establish Wilks-type results showing that the EL and AEL likelihood-ratio statistics converge to the standard ${\chi }_1^2$ distribution under DP. The resulting intervals are calibrated for the clipped population mean ${\mu }_c$, the estimand targeted by fixed-clipping bounded-sensitivity mechanisms. We explicitly characterize the clipping bias and provide a bias-aware extension for inference on the raw mean. Simulations across various distributions, together with real-data experiments, show that the proposed methods maintain coverage close to the nominal level while typically producing shorter confidence intervals than existing approaches.