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Robustness for free: asymptotic size and power of max-tests in high dimensions

Anders Bredahl Kock, David Preinerstorfer

TL;DR

The paper tackles high-dimensional mean testing under minimal moment conditions, showing that conventional max-tests based on arithmetic means can fail when $d$ grows beyond the polynomial barrier $d=o(n^{m/2-1})$. It introduces a robust max-test based on coordinatewise winsorized means, which maintains asymptotic size even when $d$ grows exponentially in $n$ and tolerates adversarial contamination, while preserving the same asymptotic power as the standard test under stronger assumptions. The authors establish high-dimensional Gaussian approximations for both uncentered and centered statistics, derive data-driven bootstrap critical values, and compare the power of robust and bootstrap-based tests, finding that bootstrap often provides gains only in specific correlation structures. Overall, the winsorized approach offers a robust, scalable alternative with comparable asymptotic performance to classical tests and explicit guidance on when bootstrap improves power.

Abstract

Consider testing a zero restriction on the mean of a $d$-dimensional random vector based on an i.i.d. sample of size $n$. Suppose further that the coordinates are only assumed to possess $m>2$ moments. Then, max-tests based on arithmetic means and critical values derived from Gaussian approximations are not guaranteed to be asymptotically valid unless $d$ is relatively small compared to $n$, because said approximation faces a polynomial growth barrier of $d=o(n^{m/2-1})$. We propose a max-test based on winsorized means, and show that it holds the desired asymptotic size even when $d$ grows at an exponential rate in $n$ and the data are adversarially contaminated. Our characterization of its asymptotic power function shows that these benefits do not come at the cost of reduced asymptotic power: the robustified max-test has identical asymptotic power to that based on arithmetic means whenever the stronger assumptions underlying the latter are satisfied. We also investigate when -- and when not -- data-driven (bootstrap) critical values can strictly increase asymptotic power of the robustified max-test.

Robustness for free: asymptotic size and power of max-tests in high dimensions

TL;DR

The paper tackles high-dimensional mean testing under minimal moment conditions, showing that conventional max-tests based on arithmetic means can fail when grows beyond the polynomial barrier . It introduces a robust max-test based on coordinatewise winsorized means, which maintains asymptotic size even when grows exponentially in and tolerates adversarial contamination, while preserving the same asymptotic power as the standard test under stronger assumptions. The authors establish high-dimensional Gaussian approximations for both uncentered and centered statistics, derive data-driven bootstrap critical values, and compare the power of robust and bootstrap-based tests, finding that bootstrap often provides gains only in specific correlation structures. Overall, the winsorized approach offers a robust, scalable alternative with comparable asymptotic performance to classical tests and explicit guidance on when bootstrap improves power.

Abstract

Consider testing a zero restriction on the mean of a -dimensional random vector based on an i.i.d. sample of size . Suppose further that the coordinates are only assumed to possess moments. Then, max-tests based on arithmetic means and critical values derived from Gaussian approximations are not guaranteed to be asymptotically valid unless is relatively small compared to , because said approximation faces a polynomial growth barrier of . We propose a max-test based on winsorized means, and show that it holds the desired asymptotic size even when grows at an exponential rate in and the data are adversarially contaminated. Our characterization of its asymptotic power function shows that these benefits do not come at the cost of reduced asymptotic power: the robustified max-test has identical asymptotic power to that based on arithmetic means whenever the stronger assumptions underlying the latter are satisfied. We also investigate when -- and when not -- data-driven (bootstrap) critical values can strictly increase asymptotic power of the robustified max-test.
Paper Structure (25 sections, 18 theorems, 159 equations)

This paper contains 25 sections, 18 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Setting and notation
  3. Tests based on normalized arithmetic means
  4. Proof strategy
  5. "Optimality" of the condition $d=n^{m/2-1}$ in Theorem \ref{['thm:sums']}
  6. Tests based on normalized winsorized means
  7. Winsorized means and tests based on these
  8. Data-driven (bootstrap) critical values
  9. Comparing the power of $\varphi_{n,W}$ and $\varphi_{n,W,B}$
  10. Proof of Theorem \ref{['thm:sums']}
  11. Auxiliary results concerning variance estimation
  12. A Gaussian approximation for standardized sums
  13. Gaussian approximation for $S_n$ when $\enVert[0]{\mu_n(P)}_\infty$ is "small"
  14. Gaussian approximation for $S_n$ when $\enVert[0]{\mu_n(P)}_\infty$ is not "small"
  15. Proof of Theorem \ref{['thm:sums']}
  16. ...and 10 more sections

Key Result

Theorem 3.1

Let $b_1\in(0,\infty)$, $b_2\in(b_1,\infty)$, and $m > 4$. Let $\alpha\in(0,1)$, and suppose that $d\geq 2$. If there exists a $\xi\in(0,1)$ such that $\frac{d}{n^{m/2-1-\xi}}\to 0$, then the following holds:

Theorems & Definitions (33)

  • Theorem 3.1
  • Theorem 4.1
  • Theorem 5.1
  • Theorem 5.2
  • Remark 5.1
  • Lemma A.1
  • proof
  • Corollary A.2
  • proof
  • Lemma A.3
  • ...and 23 more