  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.
