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Testing the dependency between two random variables is an important inferenceproblem in statistics since many statistical procedures rely on the assumptionthat the two samples are independent. To test whether two samples areindependent, a so-called HSIC (Hilbert--Schmidt Independence Criterion)-basedtest has been proposed. Its null distribution is approximated either bypermutation or a Gamma approximation. In this paper, a new HSIC-based test isproposed. Its asymptotic null and alternative distributions are established. Itis shown that the proposed test is root-n consistent. A three-cumulant matchedchi-squared approximation is adopted to approximate the null distribution ofthe test statistic. By choosing a proper reproducing kernel, the proposed testcan be applied to many different types of data including multivariate,high-dimensional, and functional data. Three simulation studies and two realdata applications show that in terms of level accuracy, power, andcomputational cost, the proposed test outperforms several existing tests formultivariate, high-dimensional, and functional data.

A fast and accurate kernel-based independence test with applications to high-dimensional and functional data