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We study a rank based univariate two-sample distribution-free test. The teststatistic is the difference between the average of between-group rank distancesand the average of within-group rank distances. This test statistic is closelyrelated to the two-sample Cram\'er-von Mises criterion. They are differentempirical versions of a same quantity for testing the equality of twopopulation distributions. Although they may be different for finite samples,they share the same expected value, variance and asymptotic properties. Theadvantage of the new rank based test over the classical one is its ease togeneralize to the multivariate case. Rather than using the empirical processapproach, we provide a different easier proof, bringing in a differentperspective and insight. In particular, we apply the H\'ajek projection andorthogonal decomposition technique in deriving the asymptotics of the proposedrank based statistic. A numerical study compares power performance of the rankformulation test with other commonly-used nonparametric tests andrecommendations on those tests are provided. Lastly, we propose a multivariateextension of the test based on the spatial rank.

A rank-based Cram\'er-von-Mises-type test for two samples