Benjamini-Hochberg Procedure for False Discovery Rate Control

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on statistics modeling, specifically regarding Multiple Hypothesis Testing.

Your Task
In testing $m$ simultaneous hypotheses with $p$-values $P_{(1)} \le P_{(2)} \le \dots \le P_{(m)}$, define the False Discovery Rate $\text{FDR} = \mathbb{E}[\text{FDP}]$, where $\text{FDP} = \frac{V}{\max(R, 1)}$ ($V$ = false rejections, $R$ = total rejections).
(1) State the Benjamini-Hochberg (BH) procedure at control level $\alpha$.
(2) Prove that under independent null test statistics, the BH procedure controls the FDR at $\frac{m_0}{m} \alpha \le \alpha$, where $m_0$ is the number of true nulls.

Constraints

• All variables are defined on a standard probability space $(\Omega, \mathcal{F}, \mathbb{P})$.
• Analytical expressions must be in closed form, and empirical simulations should converge within standard statistical tolerances.