Towards a new toolbox of statistical primitives: Optimally estimating the mean of a distribution

Abstract Mean estimation is perhaps the most fundamental building block of statistics, yet the tools we use are surprisingly brittle. For example, the conventional method of using the sample mean can be made arbitrarily bad with a single extreme data point. On the other hand, prior theoretical alternatives, such as median‐of‐means, trimmed mean and other high‐dimensional techniques, do not necessarily perform very well in practice either. In this article, we introduce a state‐of‐the‐art “swiss army knife” 1‐dimensional mean estimator based on a carefully constructed sample downweighting scheme, whose accuracy is optimal even in the constants under minimal assumptions, and is furthermore robust against data corruption. We also introduce a state‐of‐the‐art “very high‐dimensional” mean estimator with analogous accuracy guarantees. These estimators are a first step towards building a new toolbox of statistical primitives, with mathematical rigor, sharp optimal theoretical guarantees and superior empirical accuracy.