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Finite-sample properties of the trimmed mean

Roberto I. Oliveira, Paulo Orenstein, Zoraida F. Rico

TL;DR

The paper analyzes the finite-sample behavior of the $k$-trimmed mean as an estimator of the mean for i.i.d. data with finite variance and beyond. It develops a conditional-mean framework showing the trimmed mean concentrates with sub-Gaussian tails and, under stronger moments, enjoys precise Gaussian-approximation and confidence intervals even in deep tails. It also proves minimax-optimality under adversarial contamination, quantifying the trade-off between random fluctuations and contamination. The methods hinge on viewing the trimmed mean as an average of conditionally i.i.d. data given trimming endpoints, enabling Bernstein-type and self-normalized CLT techniques. Together, the results provide practical finite-sample guarantees, sharp constants, and robust performance analyses for trimmed-mean estimators in both light- and heavy-tailed and contaminated settings.

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.

Finite-sample properties of the trimmed mean

TL;DR

The paper analyzes the finite-sample behavior of the -trimmed mean as an estimator of the mean for i.i.d. data with finite variance and beyond. It develops a conditional-mean framework showing the trimmed mean concentrates with sub-Gaussian tails and, under stronger moments, enjoys precise Gaussian-approximation and confidence intervals even in deep tails. It also proves minimax-optimality under adversarial contamination, quantifying the trade-off between random fluctuations and contamination. The methods hinge on viewing the trimmed mean as an average of conditionally i.i.d. data given trimming endpoints, enabling Bernstein-type and self-normalized CLT techniques. Together, the results provide practical finite-sample guarantees, sharp constants, and robust performance analyses for trimmed-mean estimators in both light- and heavy-tailed and contaminated settings.

Abstract

The trimmed mean of scalar random variables from a distribution is the variant of the standard sample mean where the smallest and largest values in the sample are discarded for some parameter . In this paper, we look at the finite-sample properties of the trimmed mean as an estimator for the mean of . 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 for some . 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.
Paper Structure (41 sections, 26 theorems, 195 equations, 1 figure)

This paper contains 41 sections, 26 theorems, 195 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sub-Gaussian properties.
  3. Precise Gaussian approximation and confidence intervals
  4. Heavier tails and contamination
  5. Technical and conceptual contributions
  6. Additional background
  7. Background on the trimmed mean.
  8. Finite-sample bounds, sub-Gaussian estimators and related topics
  9. Adversarial contamination
  10. Organization
  11. Preliminaries
  12. General notation
  13. Probability notation and facts.
  14. Concentration and Gaussian approximation for i.i.d. sums
  15. Trimmed means: first steps
  16. ...and 26 more sections

Key Result

Theorem 1.1.1

Consider i.i.d. random variables $X_1,\dots,X_n$ with a well-defined mean $\mu$ and variance $\sigma^2<+\infty$. Take $0<x\leq \sqrt{{n}/{(\sqrt{2}+1)^2}-2}$ and consider the trimmed mean $\overline{X}_{n,k}$ with trimming parameter $k(x):=\lceil x^2/2\rceil$. Then:

Figures (1)

  • Figure 1: Violin plot for the three estimators under $t$ distributions with different parameters.

Theorems & Definitions (54)

  • Theorem 1.1.1: Proof in § \ref{['subsub:proof:allsubgaussian']}
  • Theorem 1.1.2: Proof in § \ref{['subsub:proof:sharpersubgaussian']}
  • Theorem 1.1.3: Proof in § \ref{['subsub:proof:multiplesubgaussian']}
  • Theorem 1.2.1: Proof in § \ref{['sub:proof:preciseconfidence']}
  • Corollary 1.2.2: Proof omitted
  • Remark 1.2.3
  • Theorem 1.3.1: Proof in Section \ref{['sec:proof:minimaxcontaminated']}
  • Remark 1.3.2
  • Remark 1.3.3
  • Proposition 2.2.1
  • ...and 44 more