Generalized Cramér–von Mises goodness-of-fit tests for multivariate distributions

A class of statistics for testing the goodness-of-fit for any multivariate continuous distribution is proposed. These statistics consider not only the goodness-of-fit of the joint distribution but also the goodness-of-fit of all marginal distributions, and can be regarded as generalizations of the multivariate Cramér–von Mises statistic. Simulation shows that these generalizations, using the Monte Carlo test procedure to approximate their finite-sample pp-values, are more powerful than the multivariate Kolmogorov–Smirnov statistic.