Note on High Dimensional Spatial-Sign Test for One Sample Problem

2026-01-13

AI Summary

This study addresses the null distribution of spatial sign tests under general scatter structures in high-dimensional, one-sample settings. It establishes that the normalized test statistic converges to a non-Gaussian limiting distribution—a mixture of a normal and a weighted chi-squared distribution—and leverages this insight to develop a wild bootstrap-based inference framework. The proposed method achieves asymptotically valid critical value estimation without imposing strong structural assumptions on the covariance matrix. Numerical experiments demonstrate that the bootstrap test accurately controls type I error across diverse dependence structures and high-dimensional, small-sample scenarios, substantially outperforming existing approaches.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

We revisit the null distribution of the high-dimensional spatial-sign test of Wang et al. (2015) under mild structural assumptions on the scatter matrix. We show that the standardized test statistic converges to a non-Gaussian limit, characterized as a mixture of a normal component and a weighted chi-square component. To facilitate practical implementation, we propose a wild bootstrap procedure for computing critical values and establish its asymptotic validity. Numerical experiments demonstrate that the proposed bootstrap test delivers accurate size control across a wide range of dependence settings and dimension-sample-size regimes.

Problem

• high-dimensional
• spatial-sign test
• one-sample problem
• null distribution
• non-Gaussian limit

Innovation

• high-dimensional spatial-sign test
• non-Gaussian limit
• wild bootstrap
• asymptotic validity
• scatter matrix