Randomized empirical processes by algebraic groups, and tests for weak null hypotheses

Randomization tests are based on a re-randomization of existing data to gain data-dependent critical values that lead exact hypothesis under special circumstances. However, it is not always possible re-randomize in accordance with the physical randomization from which have been obtained. As consequence, statistical typically cannot control type I error probability if available sample overly large. Still, similarly as bootstrap, can be used improve control. contrast bootstrap and permutation test theory, general asymptotic theory weak null hypotheses has yet developed for tests. It thus aim this paper provide conveniently applicable validity statistics asymptotically normal hypotheses. This also justify reliability randomization-based confidence intervals. To achieve this, present article creates link between two well-established fields mathematical statistics: empirical processes inference via algebraic groups. end, broadly conditional convergence theorem randomized observations. Random elements an group applied vectors version statistic derived. Combining variant functional delta-method suitable studentization statistic, deduced, while finite exactness property group-invariant sub-hypotheses preserved. The methodology exemplified three examples: Pearson correlation coefficient, Mann-Whitney effect right-censored paired data, competing risks analysis. practical usefulness approaches assessed through simulation studies application patients suffering diabetic retinopathy.