Privacy Preserving Inference on the Ratio of Two Gaussians Using (Weighted) Sums

The ratio of two Gaussians is useful in many contexts of statistical inference. We discuss statistically valid inference of the ratio estimator under Differential Privacy (DP). We use the delta method to derive the asymptotic distribution of the ratio estimator and use the Gaussian mechanism to provide $(\epsilon, \delta)$ privacy guarantees. Like many statistics, the quantities needed here can be re-written as functions of sums, and sums are easy to work with for many reasons. In the DP case, the sensitivity of a sum can be easily obtained. We focus on the coverage of 95\% confidence intervals (CIs). Our simulations shows that the no correction method, which ignores the noise mechanism, gives CIs that are too narrow to provide proper coverage for small samples. We propose two methods to mitigate the under-coverage issue, one based on Monte Carlo simulations and the other based on analytical correction. We show that the CIs of our methods have the right coverage with proper privacy budget. In addition, our methods can handle weighted data, where the weights are fixed and bounded.