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

If $B=\varnothing$, then for every $j$ at least one of $H_j^{\beta }$ and $H_j^{\eta }$ is true. Sincevalidity of the p-value for whichever component null is true implies $\mathbb{P}(q_j\le t)\le t$. Therefore the union bound givesThis is the Bonferroni correction for the composite intersection alternatives.