Adaptive L-tests for high dimensional independence

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.

💡 Research Summary

The paper addresses the fundamental problem of testing mutual independence among a high‑dimensional vector of random variables. Classical bivariate independence tests (Pearson, Kendall, Spearman, Hoeffding) do not scale to the multivariate setting, and existing high‑dimensional approaches fall into two families: sum‑type statistics that aggregate squared pairwise correlations (e.g., Schott, 2005) and are powerful against dense alternatives, and max‑type statistics that take the largest squared correlation (e.g., Jiang, 2004) and excel under sparse alternatives. Because the sparsity level of the true alternative is unknown in practice, adaptive procedures that combine both types have been proposed, typically by merging a sum‑type p‑value with a max‑type p‑value using methods such as the Cauchy combination test. However, these existing adaptive methods only exploit the global sum and the extreme maximum, which can be sub‑optimal for intermediate sparsity regimes.

The authors introduce a new class of “L‑tests” based on order statistics of the squared sample correlations. Let (p^* = p(p-1)/2) be the number of distinct variable pairs and denote the ordered squared correlations by (\hat\rho_{(1)}^2 \le \dots \le \hat\rho_{(p^*)}^2). For a chosen integer (k), the L‑statistic is defined as
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