Finite-Sample Symmetric Mean Estimation with Fisher Information Rate

• URL: http://arxiv.org/abs/2306.16573v1
• Date: Wed, 28 Jun 2023 21:31:46 GMT
• Title: Finite-Sample Symmetric Mean Estimation with Fisher Information Rate
• Authors: Shivam Gupta, Jasper C.H. Lee, Eric Price
• Abstract summary: Mean of an unknown variance-$sigma2$ distribution can be estimated from $n$ samples with variance $fracsigma2n$ and nearly corresponding subgaussian rate. When $f$ is known up to translation, this can be improved to $frac1nmathcal I$, where $mathcal I$ is the Fisher information of the distribution.
• Score: 15.802475232604667
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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