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