= Benjamini-Hochberg leave-one-out identity
{c}
Let $R$ be the number of rejections by the <Benjamini-Hochberg procedure> at level $\alpha$, and let $R_i$ be the rejection count after replacing $p_i$ by zero. Then $R_i\ge1$ and
$$
\frac{\mathbf1\{i\text{ rejected}\}}{R\vee1}=\frac{\mathbf1\{p_i\le\alpha R_i/m\}}{R_i}.
$$
If $i$ was rejected, lowering its <p-value> does not change any ordered <p-value> beyond rank $R$, so $R_i=R$. Conversely, if $p_i\le\alpha R_i/m$, the other $R_i-1$ rejected values together with $p_i$ satisfy the original threshold; monotonicity gives $R=R_i$. For a valid true-null <p-value> independent of the others, conditioning on $R_i$ bounds the expectation of the right side by $\alpha/m$. Summing over true nulls proves <false discovery rate> at most $m_0\alpha/m$.
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