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

A p-value $p_i$ for $H_i$ is super-uniform under that null: ${\mathbb{P}}_{H_i}(p_i\le u)\le u$ for every $u\in [0,1]$. For the Benjamini-Hochberg procedure, order $p_{(1)}\le \cdots \le p_{(m)}$, setwith $\hat{k}=0$ if the set is empty, and reject the hypotheses whose p-values are at most $p_{(\hat{k})}$.

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 $\{i\text{ rejected},R=r\}$ one has $p_i\le \alpha r/m$ and $R_i=r$, while $R_i$ is independent of $p_i$. Hence super-uniformity givesSumming over $i\in I_0$ proves that the false discovery rate is at most $|I_0|\alpha /m\le \alpha$.

Under the intersection null, every rejection is false, so the false-discovery proportion is $1_{\{R>0\}}$ and its expectation is the familywise error rate. Because the p-values are independent and exactly uniform, every super-uniform inequality in part (b) is an equality. Here $|I_0|=m$, and thereforeEquivalently, the order-statistic identity in the hint gives the same equality by induction on $m$.