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 to exact hypothesis tests under special circumstances. However, it is not always possible to re-randomize data in accordance to the physical randomization from which the data has been obained. As a consequence, most statistical tests cannot control the type I error probability. Still, similarly as the bootstrap, data re-randomization can be used to improve the type I error control. However, no general asymptotic theory under weak null hypotheses has been developed for such randomization tests yet. It is the aim of this paper to provide a conveniently applicable theory on the asymptotic validity of randomization tests with asymptotically normal test statistics. Similarly, confidence intervals will be developed. This will be achieved by creating a link between two well-established fields in mathematical statistics: empirical processes and inference based on randomization via algebraic groups. A broadly applicable conditional weak convergence theorem is developed for empirical processes that are based on randomized observations. Random elements of an algebraic group are applied to the data vectors from which the randomized version of a statistic is derived. Combining a variant of the functional delta-method with a suitable studentization of the statistic, asymptotically exact hypothesis tests is deduced, while the finite sample exactness property under group-invariant sub-hypotheses is preserved. The methodology is exemplified with: the Pearson correlation coefficient, a Mann-Whitney effect based on right-censored paired data, and a competing risks analysis. The practical usefulness of the approaches is assessed through simulation studies and an application to data from patients suffering from diabetic retinopathy.