The mean of an unknown variance-$\sigma^2$ distribution $f$ can be estimated from $n$ samples with variance $\frac{\sigma^2}{n}$ and nearly corresponding subgaussian rate. When $f$ is known up to translation, this can be improved asymptotically to $\frac{1}{n\mathcal I}$, where $\mathcal I$ is the Fisher information of the distribution. Such an improvement is not possible for general unknown $f$, but [Stone, 1975] showed that this asymptotic convergence $\textit{is}$ possible if $f$ is $\textit{symmetric}$ about its mean. Stone's bound is asymptotic, however: the $n$ required for convergence depends in an unspecified way on the distribution $f$ and failure probability $\delta$. In this paper we give finite-sample guarantees for symmetric mean estimation in terms of Fisher information. For every $f, n, \delta$ with $n > \log \frac{1}{\delta}$, we get convergence close to a subgaussian with variance $\frac{1}{n \mathcal I_r}$, where $\mathcal I_r$ is the $r$-$\textit{smoothed}$ Fisher information with smoothing radius $r$ that decays polynomially in $n$. Such a bound essentially matches the finite-sample guarantees in the known-$f$ setting.

翻译：未知方差$\sigma^2$的分布$f$的均值，可通过$n$个样本以方差$\frac{\sigma^2}{n}$及近似的次高斯速率进行估计。当$f$在平移意义下已知时，该估计可渐近改进至$\frac{1}{n\mathcal I}$，其中$\mathcal I$为分布的Fisher信息。对于一般未知的$f$，此类改进无法实现，但[Stone, 1975]指出：若$f$关于其均值$\textit{对称}$，则$\textit{确实}$存在此渐近收敛性。然而，Stone的界是渐近的：收敛所需$n$的取值以未指定的方式依赖于分布$f$与失败概率$\delta$。本文基于Fisher信息给出对称均值估计的有限样本保证。对于每个满足$n > \log \frac{1}{\delta}$的$f, n, \delta$，我们实现了接近于方差为$\frac{1}{n \mathcal I_r}$的次高斯分布的收敛性，其中$\mathcal I_r$为$r$-$\textit{平滑}$后的Fisher信息，平滑半径$r$随$n$多项式衰减。该界本质上匹配了已知$f$情形下的有限样本保证。