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:
- Distribution-free test,
- for Multivariate distributions,
- 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.