We present \textit{universal} estimators for the statistical mean, variance, and scale (in particular, the interquartile range) under pure differential privacy. These estimators are universal in the sense that they work on an arbitrary, unknown continuous distribution $\mathcal{P}$ over $\mathbb{R}$, while yielding strong utility guarantees except for ill-behaved $\mathcal{P}$. For certain distribution families like Gaussians or heavy-tailed distributions, we show that our universal estimators match or improve existing estimators, which are often specifically designed for the given family and under \textit{a priori} boundedness assumptions on the mean and variance of $\mathcal{P}$. This is the first time these boundedness assumptions are removed under pure differential privacy. The main technical tools in our development are instance-optimal empirical estimators for the mean and quantiles over the unbounded integer domain, which can be of independent interest.

翻译：我们提出了通用的统计平均值、方差和尺度（特别地，四分位距）的纯差分隐私估计器。这些估计器通常适用于任意未知连续分布$\mathcal{P}$，并且除了表现不佳的$\mathcal{P}$之外，具有较强的效用保证。对于某些分布族，如高斯分布或重尾分布，我们展示了我们的通用估计器可以与现有估计器相匹配或更优秀。现有的估计器经常是为特定的分布族设计的，并在$\mathcal{P}$的均值和方差被先验地限制的情况下成立。这是首次在纯差分隐私下消除这些有限制性的假设。我们主要开发的技术工具是关于未受限整数域上的平均值和分位数的实例最优经验估计器，这可能具有独立的学术意义。