Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 205 / 4 / b

Table of contents
- Solution b

Solution

Let

R

be the number of rejections and

I_{0}

the true-null indices. For

i \in I_{0}

, remove

p_{i}

and let

R_{i}

be the number determined by the corresponding leave-one-out step-up rule. On

{i rejected, R = r}

one has

p_{i} \leq α r / m

and

R_{i} = r

, while

R_{i}

is independent of

p_{i}

. Hence super-uniformity gives

E \frac{1 _{{i rejected}}}{R \lor 1} \leq \frac{α}{m} .

Summing over

i \in I_{0}

proves that the false discovery rate is at most

∣ I_{0} ∣ α / m \leq α

.

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