As in Procedure 9.1.1, suppose that a test of the individual hypothesis Hj is based on a test statistic Tn,j , with large values indicating evidence against the Hj . Assume >s j=1 ωj is not empty. For any subset K of {1,...,s}, let cn,K(α, P) denote an α-quantile of the distribution of maxj∈K Tn,j under P.

Concretely, cn,K(α, P) = inf{x : P{max j∈K Tn,j ≤ x} ≥ α} . (9.98)
For testing the intersection hypothesis HK, it is only required to approximate a critical value for P ∈ >
j∈K ωj . Because there may be many such P, we define cn,K(1 − α) = sup{cn,K(1 − α, P) : P ∈ 2 j∈K ωj} . (9.99)
(i) In Procedure 9.1.1, show that the choice ˆcn,K(1 − α) = cn,K(1 − α) controls the FWER, as long as (9.9) holds.
(ii) Further assume that for every subset K ⊂ {1,...,k}, there exists a distribution PK which satisfies cn,K(1 − α, P) ≤ cn,K(1 − α, PK) (9.100)
for all P such that I(P) ⊃ K. Such a PK may be referred to being least favorable among distributions P such that P ∈ >
j∈K ωj . (For example, if Hj corresponds to a parameter θj ≤ 0, then intuition suggests a least favorable configuration should correspond to θj = 0.) In addition, assume the subset pivotality condition of Westfall and Young (1993); that is, assume there exists a P0 with I(P0) = {1,...,s} such that the joint distribution of {Tn,j : j ∈ I(PK)} under PK is the same as the distribution of {Tn,j : j ∈ I(PK)} under P0. This condition says the (joint) distribution of the test statistics used for testing the hypotheses Hj , j ∈ I(PK) is unaffected by the truth or falsehood of the remaining hypotheses (and therefore we assume all hypotheses are true by calculating the distribution of the maximum under P0). Show we can use ˆcn,K(1 − α, P0) for ˆcn,K(1 − α).
(iii) Further assume the distribution of (Tn,1,...,Tn,s) under P0 is invariant under permutations (or exchangeable). Then, the critical values ˆcn,K(1 − α) can be chosen to depend only on |K|.