AbstractNonparametric tests are proposed for the equality of two unknownp‐variate distributions. Empirical probability measures are defined from samples from the two distributions and used to construct test statistics as the supremum of the absolute differences between empirical probabilities, the supremum being taken over all possible events. The test statistics are truly multivariate in not requiring the artificial ranking of multivariate observations, and they are distribution‐free in the generalp‐variate case. Asymptotic null distributions are obtained. Powers of the proposed tests and a competitor are examined by Monte Carlo techn