#
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
*X*_{1},..., X_{m},
Y_{1},..., Y_{n},
that connect points from different samples. Friedman and Rafsky
conjectured that a test of *H*_{0}:f=g
that rejects *H*_{0} for small values of
*R*_{m,n} should have power against general alternatives.
We prove that
*R*_{m,n}
is asymptotically distribution-free under
*H*_{0},
and that the multivariate two-sample test based on
*R*_{m,n}
is universally consistent.
Annals of Statistics 27, 290-298 (1999).