Estimating the mean of a real valued distribution

Paul Valiant

I revisit the basic statistical problem of estimating the mean of a real-valued distribution. I will introduce an estimator with the guarantee that "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 finite variance, including those without any higher moments. Parameterized by the sample size nn, the failure probability δδ, and the variance σ2σ2, our estimator is accurate to within σ⋅(1+o(1))2log1δn‾‾‾‾‾‾√σ⋅(1+o(1))2log⁡1δn, which is tight up to the 1+o(1)1+o(1) factor. I will contrast these results with ongoing efforts to adapt these techniques to higher dimensions. Based on joint work with Jasper Lee - https://arxiv.org/abs/2011.08384

https://www.ias.edu/video/estimating-mean-real-valued-distribution