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

, so

r \leq ∣ B ∣ + 1 = p - m_{0} + 1, p - r + 1 \geq m_{0} .

Selection through rank

r

implies

q_{τ (r)} \leq \frac{α}{p - r + 1} \leq \frac{α}{m _{0}} .

Consequently

P (B \neq \subseteq B) \leq P (min_{j \in I_{0}} q_{j} \leq \frac{α}{m _{0}}) \leq \sum_{j \in I_{0}} P (q_{j} \leq \frac{α}{m _{0}}) \leq α .

Thus the procedure controls the familywise error rate without requiring independence among the p-values.

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