Finite-sample properties of the trimmed mean

TL;DR

This paper analyzes the finite-sample statistical properties of the trimmed mean, demonstrating its sub-Gaussian concentration, tail approximation, and minimax optimality under various assumptions and contamination scenarios.

Contribution

It provides new finite-sample bounds and robustness results for the trimmed mean as an estimator of the mean, under weaker assumptions and contamination.

Findings

Trimmed mean achieves Gaussian-type concentration around the mean.

Tail probabilities of the trimmed mean closely match Gaussian tails under certain conditions.

Trimmed mean is minimax-optimal even with adversarial contamination.

Abstract

The trimmed mean of $n$ scalar random variables from a distribution $P$ is the variant of the standard sample mean where the $k$ smallest and $k$ largest values in the sample are discarded for some parameter $k$. In this paper, we look at the finite-sample properties of the trimmed mean as an estimator for the mean of $P$. Assuming finite variance, we prove that the trimmed mean is ``sub-Gaussian'' in the sense of achieving Gaussian-type concentration around the mean. Under slightly stronger assumptions, we show the left and right tails of the trimmed mean satisfy a strong ratio-type approximation by the corresponding Gaussian tail, even for very small probabilities of the order $e^{-n^c}$ for some $c>0$. In the more challenging setting of weaker moment assumptions and adversarial sample contamination, we prove that the trimmed mean is minimax-optimal up to constants.