  To quantify the dependence between two random vectors of possibly different
dimensions, we propose to rely on the properties of the 2-Wasserstein distance.
We first propose two coefficients that are based on the Wasserstein distance
between the actual distribution and a reference distribution with independent
components. The coefficients are normalized to take values between 0 and 1,
where 1 represents the maximal amount of dependence possible given the two
multivariate margins. We then make a quasi-Gaussian assumption that yields two
additional coefficients rooted in the same ideas as the first two. These
different coefficients are more amenable for distributional results and admit
attractive formulas in terms of the joint covariance or correlation matrix.
Furthermore, maximal dependence is proved to occur at the covariance matrix
with minimal von Neumann entropy given the covariance matrices of the two
multivariate margins. This result also helps us revisit the RV coefficient by
proposing a sharper normalisation. The two coefficients based on the
quasi-Gaussian approach can be estimated easily via the empirical covariance
matrix. The estimators are asymptotically normal and their asymptotic variances
are explicit functions of the covariance matrix, which can thus be estimated
consistently too. The results extend to the Gaussian copula case, in which case
the estimators are rank-based. The results are illustrated through theoretical
examples, Monte Carlo simulations, and a case study involving
electroencephalography data.
