Robustness for Free: Asymptotic Size and Power of Max-Tests in High Dimensions

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(nm/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.