Testing for independence in high dimensions based on empirical copulas

TL;DR

This paper introduces new high-dimensional independence tests based on empirical copulas and Moebius transformations, capable of detecting higher-order dependencies even when the number of variables exceeds the sample size.

Contribution

It extends empirical copula-based independence testing to high dimensions, incorporating higher-order dependency detection and providing asymptotic normality results.

Findings

Test statistics converge to the standard normal distribution.

Method effectively detects higher-order dependencies.

Simulation results demonstrate good performance.

Abstract

Testing for pairwise independence for the case where the number of variables may be of the same size or even larger than the sample size has received increasing attention in the recent years. We contribute to this branch of the literature by considering tests that allow to detect higher-order dependencies. The proposed methods are based on connecting the problem to copulas and making use of the Moebius transformation of the empirical copula process; an approach that has already been used successfully for the case where the number of variables is fixed. Based on a martingale central limit theorem, it is shown that respective test statistics converge to the standard normal distribution, allowing for straightforward definition of critical values. The results are illustrated by a Monte Carlo simulation study.