Adaptive L-tests for high dimensional independence

TL;DR

This paper introduces adaptive L-tests for high-dimensional mutual independence, combining fixed and diverging order statistics to achieve robust, powerful testing across various scenarios.

Contribution

The paper develops a new class of adaptive tests based on L-statistics for high-dimensional independence, with proven asymptotic properties and practical advantages.

Findings

Asymptotic distribution established for fixed $k$

Asymptotic normality proved for diverging $k$

Simulation shows improved test power

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.