On the multivariate two-sample problem using strong approximations of the EDF

The recently developed strong approximation methods are discussed and applied to the problem of testing whether two independent multivariate samples come from the same population and whether the components of the observations are independent. The usual Cramér-von Mises statistic, as well as one based on the difference between the sum of the two multivariate EDF's and twice the product of the marginal EDF's of one, are studied. A fairly sensitive integral statistic is also discussed. Consistency and some asymptotic power properties are explored. Emphasis is placed on explication of the strong approximation methodology.