Finite-Sample Symmetric Mean Estimation with Fisher Information Rate

TL;DR

This paper provides finite-sample guarantees for symmetric mean estimation using Fisher information, achieving near-optimal convergence rates that depend on smoothed Fisher information, applicable to any distribution with symmetry.

Contribution

The paper introduces finite-sample bounds for symmetric mean estimation based on smoothed Fisher information, extending asymptotic results to practical, finite-sample scenarios.

Findings

Achieves near-subgaussian convergence rates with finite samples.

Introduces the concept of smoothed Fisher information for finite-sample analysis.

Matches known distribution results with finite-sample guarantees.

Abstract

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.