An exact projection pursuit-based algorithm for multivariate two-sample nonparametric testing applicable to retrospective and group sequential studies

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 Cramér-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 apply one-to-one-mapping between the equality of vectors' distributions and the equality of distributions of relevant univariate linear projections. We establish a general 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. A novel MTS nonparametric procedure in the group sequential manner is proposed. 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.