For independent d-variate random variables X₁,,Xₘ with common density f and Y₁,,Yₙ with common density g, let Rm,n be the number of edges in the minimal spanning tree with vertices X₁,,Xₘ, Y₁,,Yₙ that connect points from different samples. Friedman and Rafsky conjectured that a test of H₀: f = g that rejects H₀ for small values of Rm,n should have power against general alternatives. We prove that Rm,n is asymptotically distribution-free under H₀ , and that the multivariate two-sample test based on Rm,n is universally consistent.
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Henze et al. (1999) studied this question.