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

TL;DR

This paper analyzes the null distribution of a high-dimensional spatial-sign test, revealing a non-Gaussian limit, and proposes a bootstrap method for accurate hypothesis testing in complex dependence settings.

Contribution

It characterizes the asymptotic distribution of the test statistic and introduces a valid bootstrap procedure for practical implementation.

Findings

The standardized test statistic converges to a mixture distribution.

The bootstrap method accurately controls size in various dependence scenarios.

Numerical experiments confirm the effectiveness of the proposed approach.

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.