*Bounty: 100*

*Bounty: 100*

I’m trying to understand the proof in Benjamini & Hochberg’s 1995 paper, specifically the Lemma in the appendix, as the rest of the proof is short and follows it.

I got stuck somewhere after equation (5)—where it says: "Thus all $m_0+j_0$ hypotheses are rejected"—why is that? By which procedure? Procedure (1) (=BH)? Or by using the cutoff declared earlier? (largest j satisfying $p_j le frac{m_0+j}{m+1}q^*$—which I understand is defined only for the False Null) This would only be true if we indeed fix the cutoff value, which is defined only on the False Null (i.e. discoveries) and simply reject any p-value below this. But I don’t immediately see how this is true if we use procedure (1)…

Also the first inequality of equation (6) seems to me wrong—it should be equality, as $p”$ is defined to be the value it’s replaced with…

In any case after that I completely get lost until equation (8). I have no idea how they arrive that there must be a $kle m_0+j-1$ for which $p_{(k)}le{k/(m+1)}q^*$—what is that $j$? Why $-1$?

From (8) onward it’s understood.

Benjamini, Y., & Hochberg, Y. (1995). Controlling the False Discovery Rate: A Practical and Powerful Approach to Multiple Testing. *Journal of the Royal Statistical Society*. Series B (Methodological), 57(1), 289–300.