A <p-value> $p_i$ for $H_i$ is super-uniform under that null: $\mathbb P_{H_i}(p_i\leq u)\leq u$ for every $u\in[0,1]$. For the <Benjamini-Hochberg procedure>, order $p_{(1)}\leq\cdots\leq p_{(m)}$, set
$$
\widehat k=\max\{k:p_{(k)}\leq\alpha k/m\},
$$
with $\widehat k=0$ if the set is empty, and reject the hypotheses whose p-values are at most $p_{(\widehat k)}$.

Solution

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

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Order $m$ p-values as $p_{(1)}\leq\cdots\leq p_{(m)}$ and take the largest $k$ with $p_{(k)}\leq\alpha k/m$. The Benjamini-Hochberg procedure rejects the hypotheses corresponding to $p_{(1)},\ldots,p_{(k)}$ and controls the <false discovery rate> under independence and several forms of positive dependence.

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