Dimension-agnostic inference using cross U-statistics

TL;DR

This paper develops a dimension-agnostic statistical inference method using cross U-statistics, enabling valid tests regardless of the relationship between sample size and dimensionality, and achieves minimax optimal power.

Contribution

It introduces a novel approach combining variational representations, sample splitting, and self-normalization to create a test statistic with a Gaussian limit independent of dimensionality.

Findings

The method provides valid inference for any dimension-to-sample size ratio.

It achieves minimax rate-optimal power against local alternatives.

Matches high-dimensional power of traditional U-statistics up to a factor of rac{rac{1}{2}}.

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.