Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 205 / 2 / Solution

The false discovery rate iswhere $R$ is the total number of rejections and $V$ is the number of rejected true nulls. The Benjamini-Hochberg procedure orders the p-values and takeswith $R=0$ when the set is empty, then rejects the $R$ smallest p-values.

For $i\in I_0$, apply the modified procedure to $p_{-i}$ with critical valuesand let $R_i$ be its number of rejections. If the full procedure rejects $i$ and makes $r$ rejections, then $R_i=r-1$, and converselyUnder Assumption A and the super-uniformity of a true-null p-value,Summing over $i\in I_0$ gives $\mathrm{FDR}\le \alpha m_0/m\le \alpha$.

Increasing any coordinates of $p_{-i}$ cannot increase the number of modified BH rejections. Henceis an increasing set.

Under Assumption B, put $t_r=\alpha r/m$ and $D_0=\varnothing$. The contribution of true null $i$ is bounded byNowPositive regression dependence and $t_{r-1}\le t_r$ implyThe resulting sum telescopes to at most one. Thus each true null again contributes at most $\alpha /m$, proving FDR control under Assumption B.