Asymptotic Distributions of High-Dimensional Nonparametric Inference with Distance Correlation
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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.
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