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August 2021 Asymptotic distributions of high-dimensional distance correlation inference
Lan Gao, Yingying Fan, Jinchi Lv, Qi-Man Shao
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Ann. Statist. 49(4): 1999-2020 (August 2021). DOI: 10.1214/20-AOS2024

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.

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Lan Gao. Yingying Fan. Jinchi Lv. Qi-Man Shao. "Asymptotic distributions of high-dimensional distance correlation inference." Ann. Statist. 49 (4) 1999 - 2020, August 2021. https://doi.org/10.1214/20-AOS2024

Information

Received: 1 July 2020; Revised: 1 September 2020; Published: August 2021
First available in Project Euclid: 29 September 2021

Digital Object Identifier: 10.1214/20-AOS2024

Subjects:
Primary: 62E20 , 62H20
Secondary: 62G10 , 62G20

Keywords: blockchain , central limit theorem , Distance correlation , high dimensionality , nonlinear dependence detection , nonparametric inference , power , rate of convergence , test of independence

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 4 • August 2021
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