Interpoint-ranking sign covariance for the test of independence

We generalize the sign covariance introduced by Bergsma & Dassios (2014) to multivariate random variables and beyond. The new interpoint-ranking sign covariance is applicable to general types of random objects as long as a meaningful similarity measure can be defined, and it is shown to be zero if and only if the two random variables are independent. The test statistic is a $U$-statistic, whose large-sample behaviour guarantees that the proposed test is consistent against general types of alternatives. Numerical experiments and data analyses demonstrate the superior empirical performance of the proposed method.