A multivariate extension of Azadkia-Chatterjee's rank coefficient

The Azadkia-Chatterjee coefficient is a rank-based measure of dependence between a random variable Y ∈ ℝ and a random vector boldsymbol Z ∈ ℝdZ. In this paper, we propose a multivariate extension that measures the dependence between random vectors boldsymbol Y ∈ ℝdY and boldsymbol Z ∈ ℝdZ, based on n i.i.d. samples. The proposed coefficient converges almost surely to a limit with the following properties: i) it lies in [0, 1]; ii) it is equal to zero if and only if boldsymbol Y and boldsymbol Z are independent; and iii) it is equal to one if and only if boldsymbol Y is almost surely a function of boldsymbol Z. Remarkably, the only assumption required by this convergence is that boldsymbol Y is not almost surely a constant vector. We further prove that under the same mild condition and after a proper scaling, this coefficient converges in distribution to a standard normal random variable when boldsymbol Y and boldsymbol Z are independent. This asymptotic normality result allows us to construct a Wald-type hypothesis test of independence based on this coefficient. To compute this coefficient, we propose a merge sort based algorithm that runs in O(n (log n)dY). Finally, we show that it can be used to measure the conditional dependence between boldsymbol Y and boldsymbol Z conditional on a third random vector boldsymbol X, and prove that the measure is monotonic with respect to the deviation from an independence distribution under certain model restrictions.