How does thresholding work in Benjamini-Hochberg hypothesis testing?

Question

I have a question about how exactly the Benjamini-Hochberg method for multiple hypothesis testing works. The Wikipedia page for False discovery rate says:

1. For a given $\alpha$, find the largest k such that $P_{(k)}\leq {\frac {k}{m}}\alpha$
2. Reject the null hypothesis (i.e., declare discoveries) for all $H_{(i)}$ for $i=1,\ldots ,k$

My question is, what if there's a hypothesis $H_{(i)}$ with $i < k$ but where $P_{(k)} > {\frac {k}{m}}\alpha$ ? Would this hypothesis be called significant? This is probably rare, but for instances where p-values increase with a slope less than $\frac{1}{m}$ (where $m$ is the number of hypotheses tested), it's possible I think.

Answer 1

Assuming that you are interested in if it is possible to have $i<k$ and $p_{(i)}>\frac{k}{m}\alpha$, then I would think it is impossible because the Benjamini-Hochberg first reorders all of the p-values from smallest to largest. (I am going to assume all p-values are unique)

$$ p_1,...,p_n\rightarrow p_{(1)},...,p_{(n)} $$

So, if $p_{(k)}\leq \frac{k}{m}\alpha$, this implies that for all $i<k$, $p_{(i)}<p_{(k)}<\frac{k}{m}\alpha$.