  Sum-based global tests are highly popular in multiple hypothesis testing. In
this paper we propose a general closed testing procedure for sum tests, which
provides confidence lower bounds for the proportion of true discoveries (TDP),
simultaneously over all subsets of hypotheses. Our method allows for an
exploratory approach, as simultaneity ensures control of the TDP even when the
subset of interest is selected post hoc. It adapts to the unknown joint
distribution of the data through permutation testing. Any sum test may be
employed, depending on the desired power properties. We present an iterative
shortcut for the closed testing procedure, based on the branch and bound
algorithm, which converges to the full closed testing results, often after few
iterations. Even if it is stopped early, it controls the TDP. The feasibility
of the method for high dimensional data is illustrated on brain imaging data.
We compare the properties of different choices for the sum test through
simulations.
