Optimal Sub-Gaussian Mean Estimation in $\mathbb{R}$

TL;DR

This paper introduces a new mean estimator for real-valued distributions that achieves sub-Gaussian accuracy without prior variance knowledge, applicable to distributions with bounded variance.

Contribution

It presents a novel variance-agnostic estimator with sub-Gaussian convergence, extending mean estimation techniques to broader distribution classes.

Findings

Estimator achieves near-optimal accuracy with high probability.

Framework generalizes to other problems involving dependent random variables.

Analysis employs mathematical programming and duality techniques.

Abstract

We revisit the problem of estimating the mean of a real-valued distribution, presenting a novel estimator with sub-Gaussian convergence: intuitively, "our estimator, on any distribution, is as accurate as the sample mean is for the Gaussian distribution of matching variance." Crucially, in contrast to prior works, our estimator does not require prior knowledge of the variance, and works across the entire gamut of distributions with bounded variance, including those without any higher moments. Parameterized by the sample size $n$, the failure probability $\delta$, and the variance $\sigma^2$, our estimator is accurate to within $\sigma\cdot(1+o(1))\sqrt{\frac{2\log\frac{1}{\delta}}{n}}$, tight up to the $1+o(1)$ factor. Our estimator construction and analysis gives a framework generalizable to other problems, tightly analyzing a sum of dependent random variables by viewing the sum implicitly as a 2-parameter $\psi$-estimator, and constructing bounds using mathematical programming and duality techniques.