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 $\alpha$ 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.