The Mann–Whitney–Wilcoxon rank sum test is limited to comparison of two groups with univariate responses. In this paper, we introduce a class of stochastic linear hypotheses that addresses these limitations within a nonparametric setting. We formulate hypotheses for simultaneous comparisons of several, multivariate response groups, without modelling the response distributions. Inference is developed based on U‐statistics theory and an exchangeability assumption. The latter condition is required to identify testable hypotheses for high‐dimensional response vectors, such as those arising in genomic and psychosocial research. The methodology is illustrated with two real‐data applications.