Universal Private Estimators

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 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 $\textit{priori}$ boundedness assumptions on the mean and variance of $\mathcal{P}$. The removal of these boundedness assumptions is surprising, as existing work believes that they are necessary under pure differential privacy.