Closed testing procedure

Choose a level-$\alpha$ local test for every nonempty intersection $H_I={\bigcap }_{i\in I}{H}_i$. The closed testing procedure rejects an elementary hypothesis $H_i$ exactly when every intersection hypothesis containing it is rejected by its local test.

If any true elementary hypothesis is rejected by a closed testing procedure, then the intersection of all true null hypotheses must also be rejected. Its local test has level $\alpha$, so the familywise error rate is at most $\alpha$ under arbitrary dependence.