High-Dimensional Distance Correlation Inference

The Department of Mathematics in the College of Arts and Sciences is honored to welcome Dr. Lan Gao to deliver the weekly colloquium. Dr. Gao is a postdoctoral scholar in the Data Sciences and Operations Department, Marshall School of Business, University of Southern California, where she works in high-dimensional inference, nonparametric statistics, causal inference, and machine learning.

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 talk, we will fill this 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 the null hypothesis. Our new theoretical results reveal an interesting phenomenon of blessing of dimensionality in the sense that 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. 

Contact Leah Quinones at Laquinon@syr.edu for a zoom link