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.
Categories: math.ST cs.IT cs.LG math.IT math.PR stat.ML stat.TH