Generalizing Distance Covariance to Measure and Test Multivariate Mutual Dependence
This work addresses the need for multivariate mutual dependence measures in statistics, offering incremental extensions to existing methods.
The authors tackled the problem of measuring and testing mutual dependence among multiple random vectors, proposing three measures that generalize distance covariance, with empirical tests showing effectiveness in simulations and real data.
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.