Martingalized U-statistics-based sequential inference: a blessing of degeneracy
Abstract
Degenerate degree-two U-statistics have non-Gaussian limits, making sequential inference substantially more difficult than in the nondegenerate case. However, row normalization preserves the target parameter and makes the centered numerator a martingale on a harmonic variance clock, at the cost of logarithmic variance inflation. We establish a strong Gaussian approximation for this process and consistently estimate its variance through online updates, allowing spectrum-free calibration. Combining this strong approximation with Gaussian-mixture (GM) and stitched law-of-the-iterated-logarithm (LIL) boundaries yields confidence sequences with asymptotic time-uniform coverage and corresponding sequential tests for null hypotheses under which the kernel is degenerate. We further extend the framework to multi-dimensional vector-valued kernels, yielding Mahalanobis confidence ellipsoids based on consistent covariance estimation. Numerical experiments illustrate sequential type-I error control and favorable detection power in two-sample and independence testing against spectral baselines, and in three-sample homogeneity testing using vector confidence ellipsoids.
Related
- Sequential Probability Ratio Test using Z-Statistics (SPRT-z): A Practical Approach for Online Experimentation
- Learning U-Statistics with Active Inference
- Nonparametric Inference Based on Expected Order Statistics
- High-Dimensional Two-Sample Inference via Marginal Likelihood-Ratio Rank Statistics
- Neural Estimation of Summary Statistics for Simulation-Based Inference with Non-Asymptotic Guarantees
BibTeX
@misc{li2026martingalized,
title = {{Martingalized U-statistics-based sequential inference: a blessing of degeneracy}},
author = {Weijia Li and Leheng Cai and Qirui Hu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00753},
eprint = {2611.00753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00753},
}
RIS
TY - UNPB
TI - Martingalized U-statistics-based sequential inference: a blessing of degeneracy
AU - Weijia Li
AU - Leheng Cai
AU - Qirui Hu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00753
UR - https://arxiv.org/abs/2611.00753
AB - Degenerate degree-two U-statistics have non-Gaussian limits, making sequential inference substantially more difficult than in the nondegenerate case. However, row normalization preserves the target parameter and makes the centered numerator a martingale on a harmonic variance clock, at the cost of logarithmic variance inflation. We establish a strong Gaussian approximation for this process and consistently estimate its variance through online updates, allowing spectrum-free calibration. Combining this strong approximation with Gaussian-mixture (GM) and stitched law-of-the-iterated-logarithm (LIL) boundaries yields confidence sequences with asymptotic time-uniform coverage and corresponding sequential tests for null hypotheses under which the kernel is degenerate. We further extend the framework to multi-dimensional vector-valued kernels, yielding Mahalanobis confidence ellipsoids based on consistent covariance estimation. Numerical experiments illustrate sequential type-I error control and favorable detection power in two-sample and independence testing against spectral baselines, and in three-sample homogeneity testing using vector confidence ellipsoids.
ER -