Dimension-agnostic inference using cross U-statistics

Abstract

Classical asymptotic theory for statistical inference usually involves calibrating a statistic by fixing the dimension d while letting the sample size n increase to infinity. Recently, much effort has been dedicated towards understanding how these methods behave in high-dimensional settings, where d and n both increase to infinity together. This often leads to different inference procedures, depending on the assumptions about the dimensionality, leaving the practitioner in a bind: given a dataset with 100 samples in 20 dimensions, should they calibrate by assuming $n\gg d$, or $d∕n\approx 0.2$? This paper considers the goal of dimension-agnostic inference—developing methods whose validity does not depend on any assumption on d versus n. We introduce an approach that uses variational representations of existing test statistics along with sample splitting and self-normalization to produce a refined test statistic with a Gaussian limiting distribution, regardless of how d scales with n. The resulting statistic can be viewed as a careful modification of degenerate U-statistics, dropping diagonal blocks and retaining off-diagonal blocks. We exemplify our technique for some classical problems including one-sample mean and covariance testing, and show that our tests have minimax rate-optimal power against appropriate local alternatives. In most settings, our cross U-statistic matches the high-dimensional power of the corresponding (degenerate) U-statistic up to a $\sqrt{2}$ factor.