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Adaptive L-tests for high dimensional independence

Ping Zhao, Huifang Ma

TL;DR

This work introduces an adaptive L-statistics framework for testing mutual independence among p variables in high dimensions. By defining T_k as the sum of the k largest squared sample correlations and analyzing both fixed-k and diverging-k regimes, the authors establish distinct asymptotic behaviors and prove their asymptotic independence, enabling a Cauchy-based adaptive combination T_C. The adaptive test achieves robust power across sparse to dense dependence structures and benefits from bootstrap-based null calibration for finite samples. Simulations show that T_C outperforms existing methods across a wide range of settings and distributions, with practical applicability illustrated. The approach broadens high-dimensional independence testing by blending sum-type and max-type strengths via L-statistics, while pointing to future extensions to nonlinear dependence measures.

Abstract

Testing mutual independence among multiple random variables is a fundamental problem in statistics, with wide applications in genomics, finance, and neuroscience. In this paper, we propose a new class of tests for high-dimensional mutual independence based on $L$-statistics. We establish the asymptotic distribution of the proposed test when the order parameter $k$ is fixed, and prove asymptotic normality when $k$ diverges with the dimension. Moreover, we show the asymptotic independence of the fixed-$k$ and diverging-$k$ statistics, enabling their combination through the Cauchy method. The resulting adaptive test is both theoretically justified and practically powerful across a wide range of alternatives. Simulation studies demonstrate the advantages of our method.

Adaptive L-tests for high dimensional independence

TL;DR

This work introduces an adaptive L-statistics framework for testing mutual independence among p variables in high dimensions. By defining T_k as the sum of the k largest squared sample correlations and analyzing both fixed-k and diverging-k regimes, the authors establish distinct asymptotic behaviors and prove their asymptotic independence, enabling a Cauchy-based adaptive combination T_C. The adaptive test achieves robust power across sparse to dense dependence structures and benefits from bootstrap-based null calibration for finite samples. Simulations show that T_C outperforms existing methods across a wide range of settings and distributions, with practical applicability illustrated. The approach broadens high-dimensional independence testing by blending sum-type and max-type strengths via L-statistics, while pointing to future extensions to nonlinear dependence measures.

Abstract

Testing mutual independence among multiple random variables is a fundamental problem in statistics, with wide applications in genomics, finance, and neuroscience. In this paper, we propose a new class of tests for high-dimensional mutual independence based on -statistics. We establish the asymptotic distribution of the proposed test when the order parameter is fixed, and prove asymptotic normality when diverges with the dimension. Moreover, we show the asymptotic independence of the fixed- and diverging- statistics, enabling their combination through the Cauchy method. The resulting adaptive test is both theoretically justified and practically powerful across a wide range of alternatives. Simulation studies demonstrate the advantages of our method.
Paper Structure (14 sections, 19 theorems, 183 equations, 2 figures, 1 table)

This paper contains 14 sections, 19 theorems, 183 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. L-tests
  3. Fixed parameter
  4. Diverging parameter
  5. Adaptive procedures
  6. Simulation
  7. Conclusion
  8. Appendix
  9. Technical tools
  10. Proof of Theorem \ref{['asymptotic distribution of ordered statistics']} (i)
  11. Proof of Theorem \ref{['asymptotic distribution of ordered statistics']} (ii)
  12. Proof of Theorem \ref{['bootstrap']}
  13. Proof of Theorem \ref{['asymptotic normality']}
  14. Proof of Theorem \ref{['asymptotic independence']}

Key Result

Theorem 1

Under the IC model given in model, suppose $\mathrm{E}\{\exp(t_0|Z_{11}|^{a})\}<\infty$ for some $0<a\le 2$ and $t_0>0$. Assume $p=p(n)\to \infty$ and $\log p=o(n^\beta)$ as $n\to\infty$, where $\beta=a/(4+a)$. Then as $\min(n,p)\to\infty$, we have (i) for all integer $1\le s\le p^*$ and $x\in\mathb (ii) for all integer $2\le l\le p^*$ and $x_1\ge \cdots\ge x_l\in\mathbb{R}$, where $b_p=4\log p-\

Figures (2)

  • Figure 1: Power curves of each test with $n=100,p=100$.
  • Figure 2: Power curves of each test with $n=200,p=100$.

Theorems & Definitions (40)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • Theorem 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 30 more