An exact distribution-free test comparing two multivariate distributions based on adjacency

Summary.  A new test is proposed comparing two multivariate distributions by using distances between observations. Unlike earlier tests using interpoint distances, the new test statistic has a known exact distribution and is exactly distribution free. The interpoint distances are used to construct an optimal non-bipartite matching, i.e. a matching of the observations into disjoint pairs to minimize the total distance within pairs. The cross-match statistic is the number of pairs containing one observation from the first distribution and one from the second. Distributions that are very different will exhibit few cross-matches. When comparing two discrete distributions with finite support, the test is consistent against all alternatives. The test is applied to a study of brain activation measured by functional magnetic resonance imaging during two linguistic tasks, comparing brains that are impaired by arteriovenous abnormalities with normal controls. A second exact distribution-free test is also discussed: it ranks the pairs and sums the ranks of the cross-matched pairs.