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：我们在纯微分隐私条件下提出了针对统计平均值、方差和尺度（特别是四分位间距）的\textit{通用}估计器。这些估计器是通用的，因为它们适用于任意未知连续分布$\mathcal{P}$上，除非$\mathcal{P}$不能很好地处理，否则会产生强大的实用保证。对于某些分布族，如高斯分布或重尾分布，我们证明了我们的通用估计器与现有估计器相匹配或更好，通常是专门设计用于给定的族群并在$\mathcal{P}$的均值和方差的\textit{先验}有界性假设下。这是纯微分隐私下首次消除这些有界性假设。我们开发的主要技术工具是整个未限制整数域上的实例最优经验估计器，这可能是独立感兴趣的。