= 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 $\{i\text{ rejected},R=r\}$ one has $p_i\leq\alpha r/m$ and $R_i=r$, while $R_i$ is independent of $p_i$. Hence super-uniformity gives
$$
\mathbb E\frac{\mathbf1_{\{i\text{ rejected}\}}}{R\vee1}
\leq\frac\alpha m.
$$
Summing over $i\in I_0$ proves that the <false discovery rate> is at most $|I_0|\alpha/m\leq\alpha$.
Back to article page
