Abstract We propose a new nonparametric independence test for two functional random variables. The test is based on a new dependence metric, the so-called angle covariance, which fully characterizes the independence of the random variables and generalizes the projection covariance proposed for random vectors. The angle covariance has a number of desirable properties, including the equivalence of its zero value and the independence of the two functional variables, and it can be applied to any functional data without finite moment conditions. We construct a V -statistic estimator of the angle covariance, and show that it has a Gaussian chaos limiting distribution under the independence null hypothesis and a normal limiting distribution under the alternative hypothesis. The test based on the estimated angle covariance is consistent against all alternatives and easy to be implemented by the given random permutation method. Simulations show that the test based on the angle covariance outperforms other competing tests for functional data.

中文翻译：

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通过角度协方差测试函数变量的独立性

摘要 我们为两个函数随机变量提出了一种新的非参数独立性检验。该测试基于一种新的依赖度量，即所谓的角度协方差，它充分表征了随机变量的独立性并概括了为随机向量提出的投影协方差。角度协方差具有许多理想的特性，包括其零值的等价性和两个函数变量的独立性，它可以应用于没有有限矩条件的任何函数数据。我们构造了一个角度协方差的 V 统计估计量，并表明它在独立原假设下具有高斯混沌极限分布，在备择假设下具有正态极限分布。基于估计的角度协方差的测试与所有替代方案一致，并且易于通过给定的随机排列方法实现。模拟表明，基于角度协方差的测试优于其他功能数据的竞争测试。