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

Ping Zhao, Long Feng

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.

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

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.
Paper Structure (11 sections, 10 theorems, 107 equations, 6 tables)

This paper contains 11 sections, 10 theorems, 107 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Spatial Sign based Test
  3. Limit null distribution
  4. Wild Bootstrap
  5. Simulation
  6. Conclusion
  7. Appendix
  8. Proof of Theorem \ref{['thm:universality']}
  9. Proof of Theorem \ref{['thm:gauss-reduction']}
  10. Proof of Lemma \ref{['lem:rad_to_gauss']}
  11. Proof of Theorem \ref{['thm:wild_bootstrap']}

Key Result

Theorem 2.1

Assume $\mathbb{P}(X_1=0)=0$ and $\tau=\mathrm{tr}(\mathbf{\Sigma}_U^2)>0$. Then for a universal constant $C>0$. Consequently, if $\kappa_4=o(n^{2/3})$, then

Theorems & Definitions (17)

  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5: A sufficient condition for asymptotic normality
  • Proposition 2.6: A universal bound
  • proof
  • Remark 2.7: Spherical case
  • Remark 2.8: When $\kappa_4$ is typically bounded for ACG directions
  • Remark 2.9
  • ...and 7 more