  In this paper we study the problem of measuring and testing joint
independence for a collection of multivariate random variables. Using the
emerging theory of optimal transport (OT) based multivariate ranks, we propose
a distribution-free test for multivariate joint independence. Towards this we
introduce the notion of rank joint distance covariance (RJdCov), the
higher-order rank analogue of the celebrated distance covariance measure, that
captures the dependencies among all the subsets of the variables. The RJdCov
can be easily estimated from the data without any moment assumptions and the
associated test for joint independence is universally consistent. We can
calibrate the test without any knowledge of the (unknown) marginal
distributions (due to the distribution-free property), both asymptotically and
in finite samples. In addition to being distribution-free and universally
consistent, the proposed test is also statistically efficient, that is, it has
non-trivial asymptotic (Pitman) efficiency. We demonstrate this by computing
the limiting local power of the test for both mixture alternatives and joint
Konijn alternatives. We also use the RJdCov measure to develop a method for
independent component analysis (ICA) that is easy to implement and robust to
outliers and contamination. Extensive simulations are performed to illustrate
the efficacy of the proposed test in comparison to other existing methods.
Finally, we apply the proposed test to learn the higher-order dependence
structure among different US industries based on stock prices.
