  We propose a general, modular method for significance testing of groups (or
clusters) of variables in a high-dimensional linear model. In presence of high
correlations among the covariables, due to serious problems of identifiability,
it is indispensable to focus on detecting groups of variables rather than
singletons. We propose an inference method which allows to build in
hierarchical structures. It relies on repeated sample splitting and sequential
rejection, and we prove that it asymptotically controls the familywise error
rate. It can be implemented on any collection of clusters and leads to improved
power in comparison to more standard non-sequential rejection methods. We
complete the theoretical analysis with empirical results for simulated and real
data.
