February 2024 Testing for independence in high dimensions based on empirical copulas
Axel Bücher, Cambyse Pakzad
Author Affiliations +
Ann. Statist. 52(1): 311-334 (February 2024). DOI: 10.1214/23-AOS2348


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 is related to Lancaster interactions and 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.

Funding Statement

This work has been supported by the Collaborative Research Center “Statistical modeling of nonlinear dynamic processes” (SFB 823) of the German Research Foundation, which is gratefully acknowledged. Computational infrastructure and support were provided by the Centre for Information and Media Technology at Heinrich Heine University Düsseldorf, which is gratefully acknowledged.


The authors are grateful to three unknown referees and an Associate Editor for their constructive comments on an earlier version of this paper.

Most parts of this paper were written when A. Bücher and C. Pakzad were affiliated with Heinrich Heine University Düsseldorf.


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Axel Bücher. Cambyse Pakzad. "Testing for independence in high dimensions based on empirical copulas." Ann. Statist. 52 (1) 311 - 334, February 2024. https://doi.org/10.1214/23-AOS2348


Received: 1 April 2022; Revised: 1 December 2023; Published: February 2024
First available in Project Euclid: 7 March 2024

MathSciNet: MR4718417
Digital Object Identifier: 10.1214/23-AOS2348

Primary: 62G10 , 62H15
Secondary: 60F05

Keywords: empirical copula process , High-dimensional statistics , higher-order dependence , Lancaster interaction , Moebius transform , rank based inference

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 1 • February 2024
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