BH procedure

Procedure to achieve FDR control. Each hypothesis $H_i$ has a realized p-value $p_i$. If we reject all hypotheses with the smallest $k^{\ast }$ p-values, where

$$k^{\ast }=\max \{k:\frac{K{p}_{(k)}}{k}\le \alpha \}.$$

Then

$$\mathbb{E}[F_D/|D|]\le \frac{{\ell }_K{K}_0}{K}\alpha ,$$

where $K_0$ is the number of true null hypothesis and ${\ell }_K={\sum }_{k}1/k$. If the p-values satisfy PRDS, then we can get rid of the ${\ell }_K$: $\mathbb{E}[F_D/|D|]\le \alpha K_0/K$.