DiPMInd: Distance Profile based Mutual Independence testing for random objects

TL;DR

This paper introduces a new framework for testing mutual independence among diverse random objects using joint distance profiles, achieving optimal rates and strong empirical performance.

Contribution

It develops a unified, flexible testing methodology based on joint distance profiles that generalizes existing methods and improves power and applicability.

Findings

The proposed tests are minimax rate optimal.

They are asymptotically distribution-free under certain conditions.

The tests outperform existing approaches in simulations and real data.

Abstract

This paper develops a novel unified framework for testing mutual independence among random objects residing in possibly different metric spaces. The framework generalizes existing methodologies and introduces new measures of mutual independence, and proposes associated tests that achieve minimax rate optimality and exhibit strong empirical power. The foundation of the proposed tests is the new concept of joint distance profiles, which uniquely characterize the joint law of random objects under a mild condition on either the joint law or the metric spaces. Our test statistics quantify the difference of the joint distance profiles of each data point with respect to the joint law and the product of marginal laws of the vector of random objects. To enhance power, we consider integrating this difference with respect to different measures and incorporate flexible data-adaptive weight profiles in the test statistics. We derive the limiting distribution of the test statistics under the null hypothesis of mutual independence and show that the proposed tests with certain weight profiles are asymptotically distribution-free if the marginal distance profiles are continuous. Furthermore, we establish the consistency of the tests under sequences of alternative hypotheses converging to the null. For practical implementations, we employ a permutation scheme to approximate the $p$-values and provide theoretical guarantees that the permutation-based tests maintain type I error control under the null and achieve consistency under alternatives. We demonstrate the power of the proposed tests across various types of data objects through simulations and real data applications, where the new tests exhibit better performance compared with popular existing approaches.