We obtain the asymptotic distributions of different multivariate parametric and nonparametric tests for the situation where the number of replications is limited, whereas the number of treatments goes to infinity (large k, small n case). For the parametric case, we consider the Dempster's ANOVA-type, Wilks Lambda, Lawley-Hotelling and Bartlett-Nanda-Pillai Statistics. In the nonparametric case, we propose the rank-analogs of the Dempster's ANOVA-type, Lawley-Hotelling and Bartlett-Nanda-Pillai statistics. The tests are based on separate rankings for the different variables.

We provide a finite sample approximation procedure in both the parametric and nonparametric cases. The finite performance of the tests is investigated through simulations. It turns out that the proposed nonparametric tests perform very well as compared to their parametric competitors, especially in the presence of outliers.

An example illustrates the application.