Asymptotic Distributions of High-Dimensional Distance Correlation Inference

• Activity Number: 19 - Statistical Inference for Random Networks and Matrices
• Type: Topic-Contributed
• Date/Time: Sunday, August 8, 2021 : 1:30 PM to 3:20 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Lan Gao* and Yingying Fan and Jinchi Lv and Qi-Man Shao
• Companies: University of Southern California and University of Southern California and University of Southern California and Southern University of Science and Technology and The Chinese University of Hong Kong
• Keywords: Nonparametric inference; high dimensionality; test of independence; nonlinear dependence detection; central limit theorem; power

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 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 the rescaled 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 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.