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：我们提出了在纯差分隐私下针对统计均值、方差和尺度（特别是四分位距）的\textit{通用}估计器。这些估计器是通用的，能够处理定义在实数集 $\mathbb{R}$ 上的任意未知连续分布 $\mathcal{P}$，同时为除不良行为分布 $\mathcal{P}$ 之外的情形提供强效用保证。对于高斯分布或重尾分布等特定分布族，我们证明通用估计器能够匹配或改进现有估计器，而这些现有估计器通常是为给定分布族专门设计的，并依赖于对 $\mathcal{P}$ 的均值和方差的\textit{先验}有界性假设。这是首次在纯差分隐私下消除这些有界性假设。我们开发过程中的主要技术工具是针对无界整数域的均值和分位数的实例最优经验估计器，这些工具可能具有独立的研究价值。