Boot-AVT: Asymptotically Anytime-Valid Testing with Degenerate U-Statistics via Bootstrap
Abstract
Many nonparametric tests are based on U-statistics that are degenerate under the null hypothesis. We develop Boot-AVT, a bootstrap procedure for asymptotically anytime-valid testing with these statistics. Using Brownian scaling and time inversion for the limiting Gaussian chaos process, we show that logarithmically penalized maxima over observed and future sample sizes have a common limiting null distribution. A quadratic multiplier bootstrap estimates the corresponding critical values. We also develop Boot-AVT-Multi for joint testing with multiple kernels. For both procedures, we establish uniform consistency of the bootstrap critical values, explicit bounds on size distortion, and asymptotic size α over an unbounded monitoring horizon as the monitoring start tends to infinity. Under fixed alternatives, we derive finite-horizon power bounds and establish power one with finite expected stopping times. Simulation studies evaluate finite-sample size and power in goodness-of-fit, two-sample and independence testing, and real-data analyses compare stopping times.
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BibTeX
@misc{liu2026bootavt,
title = {{Boot-AVT: Asymptotically Anytime-Valid Testing with Degenerate U-Statistics via Bootstrap}},
author = {Jixuan Liu and Long Feng and Liuhua Peng and Zhaojun Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28931},
eprint = {2610.28931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28931},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Boot-AVT: Asymptotically Anytime-Valid Testing with Degenerate U-Statistics via Bootstrap
AU - Jixuan Liu
AU - Long Feng
AU - Liuhua Peng
AU - Zhaojun Wang
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28931
UR - https://arxiv.org/abs/2610.28931
AB - Many nonparametric tests are based on U-statistics that are degenerate under the null hypothesis. We develop Boot-AVT, a bootstrap procedure for asymptotically anytime-valid testing with these statistics. Using Brownian scaling and time inversion for the limiting Gaussian chaos process, we show that logarithmically penalized maxima over observed and future sample sizes have a common limiting null distribution. A quadratic multiplier bootstrap estimates the corresponding critical values. We also develop Boot-AVT-Multi for joint testing with multiple kernels. For both procedures, we establish uniform consistency of the bootstrap critical values, explicit bounds on size distortion, and asymptotic size α over an unbounded monitoring horizon as the monitoring start tends to infinity. Under fixed alternatives, we derive finite-horizon power bounds and establish power one with finite expected stopping times. Simulation studies evaluate finite-sample size and power in goodness-of-fit, two-sample and independence testing, and real-data analyses compare stopping times.
ER -