Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.28931

Boot-AVT: Asymptotically Anytime-Valid Testing with Degenerate U-Statistics via Bootstrap

Jixuan Liu·Long Feng·Liuhua Peng·Zhaojun Wang
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.STstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Jixuan Liu, Long Feng, Liuhua Peng, Zhaojun Wang. 2026-10-20. Boot-AVT: Asymptotically Anytime-Valid Testing with Degenerate U-Statistics via Bootstrap. https://arxiv.org/abs/2610.28931

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Extreme change-point detection

We examine rules for predicting whether a point in $\mathbb{R}$ generated from a 50-50 mixture of two different probability distributions came from one distribution or the other, given limited (or no) information on the two distributions, and, as clues, one point generated randomly from each of the two distributions. We prove that nearest-neighbor prediction does better than chance when we know the two distributions are Gaussian densities without knowing their parameter values. We conjecture that this result holds for general probability distributions and, furthermore, that the nearest-neighbor rule is optimal in this setting, i.e., no other rule can do better than it if we do not know the distributions or do not know their parameters, or both.

math.ST↗
arXiv2026

High-dimensional linear regression inference via $\ell^2$ weak convergence

We prove weak convergence in a separable Hilbert space for estimators of high-dimensional regression coefficients, which yields asymptotic normality and enables direct use of standard asymptotic tools such as the continuous mapping theorem. The approach permits diverging sparsity with a growing number of small nonzero coefficients, while requiring that only finitely many have moderate magnitude. As applications, we develop a test for finitely many linear hypotheses and, via a Scheffé-type approach, simultaneous inference for infinitely many linear hypotheses, yielding both a global test and simultaneous confidence bands for the regression function. The limiting distributions are given by weighted sums of independent chi-squared variables, and plug-in critical values achieve asymptotically correct size.

math.ST↗
arXiv2026

Third and Fourth Moment Phenomena for Monochromatic Edges and Applications to Graph-Based Two-Sample Tests

Given a graph sequence $\{G_n\}_{n\in\mathbb{N}}$, denote by $T_n$ the number of monochromatic edges when the vertices of $G_n$ are independently colored red or blue with probabilities $p$ and $1-p$, respectively. We characterize the asymptotic normality of $T_n$, after centering and scaling to unit variance, in terms of subgraph (motif) counts and standardized moments. One of the highlights of our results is a {\it third moment phenomenon}: for every fixed $p\in(0,1)\backslash\{\frac12\}$, asymptotic normality holds if and only if the standardized third moment converges to zero. The more familiar {\it fourth moment condition}, which requires the standardized fourth moment to converge to $3$, gives an equivalent characterization except at the two exceptional probabilities $p=\frac{1\pm1/\sqrt3}{2}$. At these values, the standardized fourth moment converges to $3$ for every graph sequence with a diverging number of edges, even when Gaussian convergence fails. In contrast, in the balanced case $p=\frac12$, the fourth moment condition characterizes asymptotic normality, while the third moment condition is insufficient. These results identify three distinct regimes for moment characterizations of Gaussian convergence, determined by the coloring probability. We further show that the third moment condition remains sufficient in the sparse regime, where $p=p_n \to 0$. As an application, we derive the limiting null distribution of two-sample statistics based on undirected $K$-nearest-neighbor graphs in the Poissonized setting, identifying sharp growth conditions on $K$ for asymptotic normality.

math.ST↗