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Robust high-dimensional Gaussian and bootstrap approximations for trimmed sample means

Lucas Resende

Abstract

Most of the modern literature on robust mean estimation focuses on designing estimators which obtain optimal sub-Gaussian concentration bounds under minimal moment assumptions and sometimes also assuming contamination. This work looks at robustness in terms of Gaussian and bootstrap approximations, mainly in the regime where the dimension is exponential on the sample size. We show that trimmed sample means attain - under mild moment assumptions and contamination - Gaussian and bootstrap approximation bounds similar to those attained by the empirical mean under light tails. We apply our results to study the Gaussian approximation of VC-subgraph families and also to the problem of vector mean estimation under general norms, improving the bounds currently available in the literature.

Robust high-dimensional Gaussian and bootstrap approximations for trimmed sample means

Abstract

Most of the modern literature on robust mean estimation focuses on designing estimators which obtain optimal sub-Gaussian concentration bounds under minimal moment assumptions and sometimes also assuming contamination. This work looks at robustness in terms of Gaussian and bootstrap approximations, mainly in the regime where the dimension is exponential on the sample size. We show that trimmed sample means attain - under mild moment assumptions and contamination - Gaussian and bootstrap approximation bounds similar to those attained by the empirical mean under light tails. We apply our results to study the Gaussian approximation of VC-subgraph families and also to the problem of vector mean estimation under general norms, improving the bounds currently available in the literature.

Paper Structure

This paper contains 15 sections, 18 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. The problem
  3. Basic assumptions and notation
  4. Robust high-dimensional Gaussian and bootstrap approximations
  5. The limitations of the empirical mean
  6. High-dimensional Gaussian and bootstrap approximations for trimmed means
  7. Robust Gaussian approximation for VC-subgraph classes
  8. Vector mean estimation under general norms
  9. Proof ideas
  10. Useful lemmata
  11. Proofs for Section 3
  12. Gaussian approximation.
  13. Bootstrap approximations.
  14. Proofs for Section 4
  15. Proofs for Section 5

Key Result

Theorem 1

Let $\mathcal{F}$ be the family of all $d$ coordinate projections of $\mathbb{R}^d$ and let $p \in (2,\infty)$.

Theorems & Definitions (36)

  • Theorem 1: Threshold phenomenon for the Gaussian approximation for the empirical average; adapted from Theorems 2.1 and 2.2 of kock2024remark
  • Theorem 2: High-dimensional Gaussian approximation for trimmed means, proof in §\ref{['sec:proofideas']}
  • Theorem 3: High-dimensional bootstrap approximations for trimmed means, proof in §\ref{['sec:proofideas']}
  • Remark 1
  • Definition 1: VC-subgraph class
  • Definition 2: VC-type class
  • Theorem 4: Gaussian approximation for empirical processes, proof in §\ref{['sec:proofsec4']}
  • Remark 2
  • Theorem 5: Theorem 2.1 of chernozhukov2016empirical
  • Remark 3
  • ...and 26 more