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.

翻译：两个高斯人的比例在统计推论的许多背景下是有用的。 我们讨论在统计推论的许多情况下, 统计上有效地推断了不同隐私(DP)下的比例估计值。 我们使用三角洲方法来得出比例估计值的无症状分布, 并使用高斯机制来提供$( epsilon,\delta) 隐私保障。 与许多统计数据一样, 这里需要的数量可以重写为总金额的函数, 并且总金额很容易工作, 原因很多。 在DP案中, 一笔金额的敏感度可以很容易获得。 我们侧重于95 ⁇ 信任间隔( CIs) 。 我们的模拟显示, 没有任何纠正方法, 忽略了噪音机制, 使CIs太窄, 无法为小样本提供适当覆盖。 我们建议了两种方法来缓解覆盖不足的问题, 一种基于蒙特卡洛 模拟, 另一种基于分析更正。 我们显示, 我们方法的CIs具有适当的隐私预算的正确覆盖范围。 此外, 我们的方法可以处理加权数据, 其重量是固定和约束的。