Resampling-based inference for high-dimensional regression

We propose a novel procedure for resampling-based multiple testing in high-dimensional regression. First, we construct permutation test statistics for each individual hypothesis by means of repeated random splits of the data. In each split, half of the observations is used to perform variable selection, and half to build test statistics for the selected variables. Then we define an asymptotically exact test for any subset of hypotheses by aggregating the individual statistics through a suitable function, e.g., maximum or weighted sums. The procedure is flexible, allowing different selection techniques and combining functions. It can be embedded into closed testing methods to make simultaneous confidence statements on the proportion of true discoveries (TDP) of all subsets, valid even under post-hoc selection.