Understanding the nonlinear association between a pair of potentially high-dimensional random vectors is encountered frequently in many contemporary big data applications. Distance correlation has become an increasingly popular tool for such a purpose. Most existing works have explored its asymptotic distributions under the independence assumption when only the sample size or the dimensionality diverges. Yet its asymptotic theory for the more realistic setting when both sample size and dimensionality diverge remains largely unexplored. In this paper, we fill such a gap and establish the central limit theorems and the 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 of independence between the two random vectors. Our new theoretical results reveal an interesting phenomenon of blessing of dimensionality for high-dimensional nonparametric inference with distance correlation in the sense that the accuracy of normal approximation can increase with dimensionality. The finite-sample performance and advantages of the test statistic are illustrated with several simulation examples and a blockchain application. This is a joint work with Lan Gao and Qiman Shao.
Short bio: Jinchi Lv is Kenneth King Stonier Chair in Business Administration and Professor in Data Sciences and Operations Department of the Marshall School of Business at the University of Southern California, Professor in Department of Mathematics at USC, and an Associate Fellow of USC Dornsife Institute for New Economic Thinking (INET). He received his Ph.D. in Mathematics from Princeton University in 2007. He was McAlister Associate Professor in Business Administration at USC from 2016-2019. His research interests include statistics, machine learning, data science, business applications, and artificial intelligence and blockchain.
His papers have been published in journals in statistics, economics, computer science, information theory, and biology, and one of them was published as a Discussion Paper in Journal of the Royal Statistical Society Series B (2008). He is the recipient of Fellow of Institute of Mathematical Statistics (2019), USC Marshall Dean's Award for Research Impact (2017), Adobe Data Science Research Award (2017), the Royal Statistical Society Guy Medal in Bronze (2015), NSF Faculty Early Career Development (CAREER) Award (2010), USC Marshall Dean's Award for Research Excellence (2009), and Zumberge Individual Award from USC's James H. Zumberge Faculty Research and Innovation Fund (2008). He has served as an associate editor of the Annals of Statistics (2013-2018), Journal of Business & Economic Statistics (2018-present), and Statistica Sinica (2008-2016).