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In this paper we study the problem of measuring and testing jointindependence for a collection of multivariate random variables. Using theemerging theory of optimal transport (OT) based multivariate ranks, we proposea distribution-free test for multivariate joint independence. Towards this weintroduce the notion of rank joint distance covariance (RJdCov), thehigher-order rank analogue of the celebrated distance covariance measure, thatcaptures the dependencies among all the subsets of the variables. The RJdCovcan be easily estimated from the data without any moment assumptions and theassociated test for joint independence is universally consistent. We cancalibrate the test without any knowledge of the (unknown) marginaldistributions (due to the distribution-free property), both asymptotically andin finite samples. In addition to being distribution-free and universallyconsistent, the proposed test is also statistically efficient, that is, it hasnon-trivial asymptotic (Pitman) efficiency. We demonstrate this by computingthe limiting local power of the test for both mixture alternatives and jointKonijn alternatives. We also use the RJdCov measure to develop a method forindependent component analysis (ICA) that is easy to implement and robust tooutliers and contamination. Extensive simulations are performed to illustratethe efficacy of the proposed test in comparison to other existing methods.Finally, we apply the proposed test to learn the higher-order dependencestructure among different US industries based on stock prices.

Distribution-free joint independence testing and robust independent component analysis using optimal transport