Benjamini-Hochberg procedure

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

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 $R_i\ge 1$ andIf $i$ 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 $m_0\alpha /m$.

Replace two p-values $P_i,P_j$ by zeros in the Benjamini-Hochberg procedure at level $\alpha$, keeping the original denominator $m$, and let $R^{(ij)}$ be the resulting rejection count. Equivalently, remove the two values and apply the step-up critical values $\alpha (k+2)/m$ to the remaining ordered p-values; their rejection count is $R_{-ij}=R^{(ij)}-2$. For $r\ge 2$,If both original values are rejected, lowering them changes no ordered rank greater than $r$, so the rejection count stays $r$. Conversely, if the modified count is $r$ and both original values meet the rank-$r$ threshold, reinserting them leaves at least $r$ qualifying values. Monotonicity under lowering gives at most $r$ original rejections. This identity is deterministic and does not require independence or a distributional assumption.

For independent p-values, with $m_0$ true-null values independent and uniform on $[0,1]$, let $R^{(i)}$ denote the Benjamini-Hochberg procedure rejection count after replacing one true-null value by zero. The Benjamini-Hochberg leave-one-out identity givesFor distinct true-null indices, the Benjamini-Hochberg leave-two-out identity givesIndeed, conditional on the remaining values and $R^{(ij)}=r$, the two independent uniform values both meet their threshold with probability $(\alpha r/m{)}^2$, cancelling the denominator $r^2$. Expanding the square of the false rejection count into its diagonal terms and ordered distinct pairs, and using symmetry among the true-null values, provesIf no null is true, both sides are zero; if exactly one is true, the pair contribution is absent. This formula complements the exact false discovery rate $\alpha m_0/m$ by describing second-moment variability.

For the Benjamini-Hochberg procedure, replace a true-null p-value $P_i$ by zero and call the new rejection count $R^{(i)}$. The Benjamini-Hochberg leave-one-out identity identifies rejection with $R=r$ with the event $P_i\le \alpha r/m$ and $R^{(i)}=r$. If $P_i$ is uniform and independent of all the other p-values, each possible rejection count contributes $\alpha \mathbb{P}(R^{(i)}=r)/m$ to its expected false-discovery fraction. Summation gives $\alpha /m$ per true null and the displayed exact false discovery rate. Super-uniform independent nulls yield the corresponding upper bound.