Title: Generalizing Distance Covariance to Measure and Test Multivariate Mutual Dependence,
Abstract: We propose three measures of mutual dependence between multiple random
vectors. All the measures are zero if and only if the random vectors are
mutually independent. The first measure generalizes distance covariance from
pairwise dependence to mutual dependence, while the other two measures are sums
of squared distance covariance. All the measures share similar properties and
asymptotic distributions to distance covariance, and capture non-linear and
non-monotone mutual dependence between the random vectors. Inspired by complete
and incomplete V-statistics, we define the empirical measures and simplified
empirical measures as a trade-off between the complexity and power when testing
mutual independence. Implementation of the tests is demonstrated by both
simulation results and real data examples.