ON THE MULTIVARIATE RUNS TEST

By Norbert Henze and Mathew D. Penrose.

Suppose there are n independent d-variate random varaibles X_i with common density f, and n independent d-variate random varaibles Y_j with common density g. Let R_m,n be the number of edges in the minimal spanning tree with vertices X₁,..., X_m, Y₁,..., Y_n, that connect points from different samples. Friedman and Rafsky conjectured that a test of H₀:f=g that rejects H₀ for small values of R_m,n should have power against general alternatives. We prove that R_m,n is asymptotically distribution-free under H₀, and that the multivariate two-sample test based on R_m,n is universally consistent.

Annals of Statistics 27, 290-298 (1999).