Asymptotic Distributions of High-Dimensional Distance Correlation Inference

TL;DR

This paper develops new asymptotic distribution results for high-dimensional distance correlation, revealing a dimensionality blessing effect and enabling better dependence detection in complex, high-dimensional data.

Contribution

It provides the first comprehensive CLTs for the null distribution of high-dimensional distance correlation when both sample size and dimension diverge.

Findings

Normal approximation accuracy improves with increasing dimension.

The rescaled distance correlation effectively captures nonlinear dependence.

Simulation and blockchain data validate theoretical results.

Abstract

Distance correlation has become an increasingly popular tool for detecting the nonlinear dependence between a pair of potentially high-dimensional random vectors. Most existing works have explored its asymptotic distributions under the null hypothesis of independence between the two random vectors when only the sample size or the dimensionality diverges. Yet its asymptotic null distribution for the more realistic setting when both sample size and dimensionality diverge in the full range remains largely underdeveloped. In this paper, we fill such a gap and develop central limit theorems and associated rates of convergence for a rescaled test statistic based on the bias-corrected distance correlation in high dimensions under some mild regularity conditions and the null hypothesis. Our new theoretical results reveal an interesting phenomenon of blessing of dimensionality for high-dimensional distance correlation inference in the sense that the accuracy of normal approximation can increase with dimensionality. Moreover, we provide a general theory on the power analysis under the alternative hypothesis of dependence, and further justify the capability of the rescaled distance correlation in capturing the pure nonlinear dependency under moderately high dimensionality for a certain type of alternative hypothesis. The theoretical results and finite-sample performance of the rescaled statistic are illustrated with several simulation examples and a blockchain application.