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<p>The Hilbert-Schmidt Independence Criteria is a well-known method for quantifying the dependence between two random vectors. However, it suffers from the curse of dimensionality. In this paper, we introduce a novel nonparametric independence test specifically designed for two functional random variables <i>X</i> and <i>Y</i>. The test is based on a new dependence metric, the so-called Projection Hilbert-Schmidt Covariance (PHSC), which efficiently characterizes the dependence of the random variables and improves upon the Hilbert-Schmidt Independence Criterion. The projection Hilbert-Schmidt covariance exhibits several favorable properties. It equals zero if and only if <i>X</i> and <i>Y</i> are independent, provided that the employed kernel is characteristic. It can be applied to random elements without finite moments when the employed kernel is bounded. It admits an U-statistic estimator, which facilitates the construction of our test. We construct a test based on the estimator and provide a theoretical critical value depending on the sample size and the bandwidths. Our analysis theoretically demonstrates the probabilities of the test committing two types of errors respectively, and proves its consistency under the local alternative hypothesis. Simulations and real data analysis show that the test based on the projection Hilbert-Schmidt covariance outperforms other competing tests for functional data.</p>

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Measuring dependence between functional data via Projection Hilbert-Schmidt Covariance

  • 期刊论文

  • 发表日期：

    2025年9月22日

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  • Zhentao Tian,
  • Darong Wang,
  • Zhongzhan Zhang

摘要

The Hilbert-Schmidt Independence Criteria is a well-known method for quantifying the dependence between two random vectors. However, it suffers from the curse of dimensionality. In this paper, we introduce a novel nonparametric independence test specifically designed for two functional random variables X and Y. The test is based on a new dependence metric, the so-called Projection Hilbert-Schmidt Covariance (PHSC), which efficiently characterizes the dependence of the random variables and improves upon the Hilbert-Schmidt Independence Criterion. The projection Hilbert-Schmidt covariance exhibits several favorable properties. It equals zero if and only if X and Y are independent, provided that the employed kernel is characteristic. It can be applied to random elements without finite moments when the employed kernel is bounded. It admits an U-statistic estimator, which facilitates the construction of our test. We construct a test based on the estimator and provide a theoretical critical value depending on the sample size and the bandwidths. Our analysis theoretically demonstrates the probabilities of the test committing two types of errors respectively, and proves its consistency under the local alternative hypothesis. Simulations and real data analysis show that the test based on the projection Hilbert-Schmidt covariance outperforms other competing tests for functional data.