  Nonparametric tests for equality of multivariate distributions are frequently
desired in research. It is commonly required that test-procedures based on
relatively small samples of vectors accurately control the corresponding Type I
Error (TIE) rates. Often, in the multivariate testing, extensions of
null-distribution-free univariate methods, e.g., Kolmogorov-Smirnov and
Cramer-von Mises type schemes, are not exact, since their null distributions
depend on underlying data distributions. The present paper extends the
density-based empirical likelihood technique in order to nonparametrically
approximate the most powerful test for the multivariate two-sample (MTS)
problem, yielding an exact finite-sample test statistic. We rigorously
establish and apply one-to-one-mapping between the equality of vectors
distributions and the equality of distributions of relevant univariate linear
projections. In this framework, we prove an algorithm that simplifies the use
of projection pursuit, employing only a few of the infinitely many linear
combinations of observed vectors components. The displayed distribution-free
strategy is employed in retrospective and group sequential manners. The
asymptotic consistency of the proposed technique is shown. Monte Carlo studies
demonstrate that the proposed procedures exhibit extremely high and stable
power characteristics across a variety of settings. Supplementary materials for
this article are available online.
