# Finite-Sample Symmetric Mean Estimation with Fisher Information Rate

arXiv: https://arxiv.org/abs/2306.16573 · HTML: https://mlsanity.com/paper/2306.16573/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 35 references that are in the map.
- First posted on arXiv 28 Jun 2023.

## Builds on (most cited first, 13 of 13)

- [Sub-Gaussian estimators of the mean of a random vector](https://mlsanity.com/paper/1702.00482/) (2017)
- [Instance-optimal Mean Estimation Under Differential Privacy](https://mlsanity.com/paper/2106.00463/) (2021)
- [Mean Estimation with Sub-Gaussian Rates in Polynomial Time](https://mlsanity.com/paper/1809.07425/) (2018)
- [Efficient Statistics, in High Dimensions, from Truncated Samples](https://mlsanity.com/paper/1809.03986/) (2018)
- [Near Instance-Optimality in Differential Privacy](https://mlsanity.com/paper/2005.10630/) (2020)
- [Fast Mean Estimation with Sub-Gaussian Rates](https://mlsanity.com/paper/1902.01998/) (2019)
- [Finite-Sample Maximum Likelihood Estimation of Location](https://mlsanity.com/paper/2206.02348/) (2022)
- [Efficient Profile Maximum Likelihood for Universal Symmetric Property Estimation](https://mlsanity.com/paper/1905.08448/) (2019)
- [The Broad Optimality of Profile Maximum Likelihood](https://mlsanity.com/paper/1906.03794/) (2019)
- [High-dimensional Location Estimation via Norm Concentration for Subgamma Vectors](https://mlsanity.com/paper/2302.02497/) (2023)
- [Approximate Profile Maximum Likelihood](https://mlsanity.com/paper/1712.07177/) (2017)
- [Information Theoretic Limits of Cardinality Estimation: Fisher Meets Shannon](https://mlsanity.com/paper/2007.08051/) (2020)
- [Instance Based Approximations to Profile Maximum Likelihood](https://mlsanity.com/paper/2011.02761/) (2020)

## Built on by (most cited first, 5 of 5)

- [Large Language Models](https://mlsanity.com/paper/2307.05782/) (2023)
- [Faster Diffusion Sampling with Randomized Midpoints: Sequential and Parallel](https://mlsanity.com/paper/2406.00924/) (2024)
- [Optimality in Mean Estimation: Beyond Worst-Case, Beyond Sub-Gaussian, and Beyond 1+α Moments](https://mlsanity.com/paper/2311.12784/) (2023)
- [Performance of the empirical median for location estimation in heteroscedastic settings](https://mlsanity.com/paper/2501.16956/) (2025)
- [Instance-Optimal Adaptive Location Estimation via Multiscale Mid-Summaries](https://mlsanity.com/paper/2609.20749/) (2026)

## Abstract (arXiv)

The mean of an unknown variance-σ2 distribution f can be estimated from n samples with variance fracσ2n and nearly corresponding subgaussian rate. When f is known up to translation, this can be improved asymptotically to frac1nmathcal 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 textitis possible if f is textitsymmetric 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 δ. In this paper we give finite-sample guarantees for symmetric mean estimation in terms of Fisher information. For every f, n, δ with n > log frac1δ, we get convergence close to a subgaussian with variance frac1n mathcal Ir, where mathcal Ir is the r-textitsmoothed 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.