🤖 AI Summary
This study addresses the problem of constructing anytime-valid confidence sequences for second-order U-statistics under continuous monitoring, proposing distinct approaches for non-degenerate and degenerate cases. In the non-degenerate setting, the authors leverage Hoeffding’s projection to reduce the problem to a time-uniform central limit theorem for first-order partial sums and employ a leave-one-out jackknife estimator to yield a data-driven confidence sequence. For the more challenging degenerate case, they introduce the first anytime-valid confidence sequence by developing the SAGE boundary and a computationally feasible scheme based on truncated spectral estimation. The resulting sequences achieve near-optimal width rates of √(log log n / n) and log log n / n in the non-degenerate and degenerate regimes, respectively, with numerical experiments confirming their empirical validity and efficiency.
📝 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.