# 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.

In machine learning, the most mature one may be the kernel method A Kernel Two-Sample Test

In statistics, as you found, the most mature one may be the one based on the energy distance.

Even though the kernel people published (in 2013 Annals of stat) to claim their kernel method is somehow equivalent with the distance method.

• It doesn't look like these options account for the one-to-one mapping situation, and I was wondering if such mapping (or transformation) would have effect on the result. – MTI Mar 21 '16 at 15:24