# Distribution-free test for two-sample multivariate distributions

Suppose that $$X_1, \cdots, X_n$$ and $$Y_1, \cdots, Y_n$$ are samples of $$R^d$$ vectors with distributions $$X$$ and $$Y$$ respectively. In addition, assume that there is one-to-one mapping between the first and second samples, i.e. $$X_i \rightarrow Y_i$$.

Since I don't know the underlying distribution of the two samples, I am trying to look for a statistical test with the following characteristics:

1. Distribution-free test,
2. for Multivariate distributions,
3. preferably takes into consideration the mapping mentioned above (bipartite?)

Progress so far: I excluded K-S test (and similar ones) since I read it doesn't generalize to the multivariate case. I also excluded distance measures like K-L divergence since I read it can be unbounded, and I wanted a range from 0 to 1.

I found a paper that addresses the first two points, but not the third. It is (both seem to propose the same test):

Székely, Gábor J., and Maria L. Rizzo. "Testing for equal distributions in high dimension." InterStat 5 (2004): 1-6.

Baringhaus, L., and C. Franz. "On a new multivariate two-sample test." Journal of multivariate analysis 88.1 (2004): 190-206.

I am wondering if there is a test that also takes into consideration the mapping mentioned above.

Note: Except for the mapping problem, this question addresses the same problem as this other cross-validated question.