  This paper introduces and investigates the utilization of maximum and average
distance correlations for multivariate independence testing. We characterize
their consistency properties in high-dimensional settings with respect to the
number of marginally dependent dimensions, assess the advantages of each test
statistic, examine their respective null distributions, and present a fast
chi-square-based testing procedure. The resulting tests are non-parametric and
applicable to both Euclidean distance and the Gaussian kernel as the underlying
metric. To better understand the practical use cases of the proposed tests, we
evaluate the empirical performance of the maximum distance correlation, average
distance correlation, and the original distance correlation across various
multivariate dependence scenarios, as well as conduct a real data experiment to
test the presence of various cancer types and peptide levels in human plasma.
