Second moment of the Benjamini-Hochberg false discovery proportion ($\mathbb{E}({\mathrm{FDP}}^2)$)

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.