On the multivariate runs test

Abstract

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.