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ASYMPTOTIC DISTRIBUTIONS OF HIGH-DIMENSIONAL DISTANCE CORRELATION INFERENCE.

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Summary

This study develops new methods for high-dimensional distance correlation, revealing a "blessing of dimensionality" where accuracy improves with more data. The findings enhance nonlinear dependence detection in complex datasets.

Keywords:
62G2062H20Nonparametric inferencePrimary 62E20blockchaincentral limit theoremdistance correlationhigh dimensionalitynonlinear dependence detectionpowerrate of convergencesecondary 62G10test of independence

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Area of Science:

  • Statistics
  • Machine Learning
  • Data Science

Background:

  • Distance correlation is vital for detecting nonlinear dependencies between high-dimensional random vectors.
  • Existing research on asymptotic distributions is limited, especially when both sample size and dimensionality increase.

Purpose of the Study:

  • To develop central limit theorems for high-dimensional distance correlation under diverging sample size and dimensionality.
  • To analyze the power of a rescaled test statistic for detecting nonlinear dependence.

Main Methods:

  • Developed central limit theorems and convergence rates for a rescaled, bias-corrected distance correlation statistic.
  • Conducted power analysis under alternative hypotheses of dependence.
  • Utilized simulation examples and a blockchain application for validation.

Main Results:

  • Established theoretical results for high-dimensional distance correlation inference.
  • Demonstrated a
  • blessing of dimensionality
  • phenomenon where normal approximation accuracy increases with dimensionality.
  • Justified the rescaled statistic's capability in capturing nonlinear dependency.

Conclusions:

  • The developed methods provide robust tools for analyzing nonlinear dependence in high-dimensional data.
  • The "blessing of dimensionality" offers new insights into statistical inference.
  • The rescaled distance correlation is effective for detecting pure nonlinear relationships.