Rank-Based Tests for Mutual Independence of High-Dimensional Random Vectors via $L_q$ Norm

2026-05-24

AI Summary

This study addresses the problem of testing independence among components of high-dimensional random vectors. The authors propose a rank-based max-sum framework that constructs finite $L_q$ norm sum statistics ($q = 2, 4, 6, \infty$), tailored to both sparse and dense alternative hypotheses. They establish the asymptotic independence between the $L_q$ and $L_\infty$ statistics, enabling effective combination of evidence across different norms. By employing the Cauchy combination method to aggregate p-values derived from multiple norms, the approach achieves robust joint inference. Theoretical analysis and extensive simulations demonstrate that the proposed method exhibits strong robustness and high empirical power across various high-dimensional settings, particularly in detecting alternatives with diverse sparsity structures.

Abstract

We consider the problem of testing mutual independence among the components of a high-dimensional random vector. Building on the rank-based max-sum framework, we introduce fixed finite-$L_q$ power-sum statistics under three general classes of rank-based correlations: simple linear rank statistics, non-degenerate rank-based U-statistics and degenerate rank-based U-statistics. The proposed statistics interpolate between the dense-alternative sensitivity of the $L_2$ statistic and the sparse-alternative sensitivity of the $L_\infty$ statistic. We establish the asymptotic independence between any fixed finite-$L_q$ block and the corresponding $L_\infty$ statistic, and combine $L_2,L_4,L_6$ and $L_\infty$ p-values through a Cauchy rule. Numerical studies show that the resulting $L_{2,4,6,\infty}$ procedure is highly robust to the sparsity of the alternative and has strong empirical power across the considered designs.

Problem

• mutual independence
• high-dimensional random vectors
• rank-based tests
• L_q norm
• independence testing

Innovation

• rank-based statistics
• mutual independence testing
• high-dimensional data
• L_q norm
• Cauchy combination