  Hierarchical inference in (generalized) regression problems is powerful for
finding significant groups or even single covariates, especially in
high-dimensional settings where identifiability of the entire regression
parameter vector may be ill-posed. The general method proceeds in a fully
data-driven and adaptive way from large to small groups or singletons of
covariates, depending on the signal strength and the correlation structure of
the design matrix. We propose a novel hierarchical multiple testing adjustment
that can be used in combination with any significance test for a group of
covariates to perform hierarchical inference. Our adjustment passes on the
significance level of certain hypotheses that could not be rejected and is
shown to guarantee strong control of the familywise error rate. Our method is
at least as powerful as a so-called depth-wise hierarchical Bonferroni
adjustment. It provides a substantial gain in power over other previously
proposed inheritance hierarchical procedures if the underlying alternative
hypotheses occur sparsely along a few branches in the tree-structured
hierarchy.
