Optimal Sub-Gaussian Mean Estimation in $\mathbb{R}$
This provides a robust estimator for statistical inference across all distributions with bounded variance, including those lacking higher moments, which is incremental over prior works that required variance knowledge.
The paper tackles the problem of estimating the mean of a real-valued distribution without prior knowledge of variance, achieving sub-Gaussian convergence with accuracy within σ·(1+o(1))√(2log(1/δ)/n), tight up to the 1+o(1) factor.
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 $δ$, and the variance $σ^2$, our estimator is accurate to within $σ\cdot(1+o(1))\sqrt{\frac{2\log\frac{1}δ}{n}}$, 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 $ψ$-estimator, and constructing bounds using mathematical programming and duality techniques.