Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 205 / 4 / d / Solution

Let $I_0=B^c$ and $m_0=|I_0|$. Each $q_j$ with $j\in I_0$ is a valid p-value by the argument in part c. If the Holm step-down procedure selects any index from $I_0$, let $r$ be the rank of the first such index. All $r-1$ earlier selections belong to $B$, soSelection through rank $r$ impliesConsequentlyThus the procedure controls the familywise error rate without requiring independence among the p-values.