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

TL;DR

This paper introduces a robust max-test for high-dimensional mean testing that maintains size and power under adversarial contamination and heavy tails, outperforming standard tests.

Contribution

It proposes a quantile-winsorized max-test that is robust to contamination, requiring minimal moments, and achieves asymptotic power equivalent to standard tests under stronger conditions.

Findings

The robust test controls size under adversarial contamination.

It has asymptotic power identical to standard max-tests under stronger conditions.

Bootstrap critical values can improve power in highly correlated designs.

Abstract

Allowing for adversarial contamination and heavy tails, we study testing whether the mean of a high-dimensional random vector equals zero. Because standard max-tests based on sample averages are highly non-robust, we propose a max-test based on quantile-winsorized observations. The test controls asymptotic size under adversarial contamination and only requires $m>2$ moments, while allowing dimension to grow exponentially with sample size. We fully characterize its asymptotic power function. Comparing with the standard max-test, for which we also derive a power characterization as a benchmark, we show that robustness is obtained for free: under the stronger conditions that make the standard max-test valid, our robust test has identical asymptotic power. We also study the role of bootstrap critical values, showing that their use never decreases power, can strictly improve asymptotic power in extremely correlated designs, but often has no first-order asymptotic effect.