  We consider the class of all multiple testing methods controlling tail
probabilities of the false discovery proportion, either for one random set or
simultaneously for many such sets. This class encompasses methods controlling
familywise error rate, generalized familywise error rate, false discovery
exceedance, joint error rate, simultaneous control of all false discovery
proportions, and others, as well as seemingly unrelated methods such as gene
set testing in genomics and cluster inference methods in neuroimaging. We show
that all such methods are either equivalent to a closed testing method, or are
uniformly improved by one. Moreover, we show that a closed testing method is
admissible as a method controlling tail probabilities of false discovery
proportions if and only if all its local tests are admissible. This implies
that, when designing such methods, it is sufficient to restrict attention to
closed testing methods only. We demonstrate the practical usefulness of this
design principle by constructing a uniform improvement of a recently proposed
method.
