Asymptotic Anytime-Valid Inference for U-statistics

TL;DR

This paper develops asymptotic confidence sequences for degree-two U-statistics under continuous monitoring, addressing both nondegenerate and degenerate cases with novel theoretical tools and data-driven procedures.

Contribution

It introduces a new framework for anytime-valid inference on U-statistics, including the SAGE boundary for degenerate cases and fully data-driven confidence sequences.

Findings

Confidence sequences with asymptotic coverage guarantees are constructed.

Widths of confidence sequences match optimal time-uniform rates.

Numerical experiments support the theoretical results.

Abstract

We study asymptotic anytime-valid confidence sequences for degree-two U-statistics under continuous monitoring. In the nondegenerate case, Hoeffding's projection reduces the problem to a time-uniform central limit theory for the partial sums of the first-order projection, while the canonical remainder is shown to be negligible under mild moment assumptions. A leave-one-out jackknife estimator then yields a fully data-driven procedure, leading to confidence sequences with asymptotic coverage guarantee for the parameter of interest. In the degenerate case, we show that the U-statistic is approximated by a centered quadratic Gaussian-chaos rather than by a simple Gaussian, which poses significant challenges for sequential inference. To address this issue, we novelly develop the Spectrally Allocated Gaussian-chaos Excursion (SAGE) boundary, and then provide plug-in implementations based on truncated spectrum estimation with consistency guarantees. The resulting widths can attain the expected time-uniform optimal rates: $\sqrt{\log\log n/n}$ in the nondegenerate regime and $\log\log n/n$ in the degenerate regime. Several widely used U-statistics are discussed within the proposed framework, and numerical experiments further support the validity of the derived theory.