e-BH procedure

The e-BH procedure is the equivalent of the BH procedure for FDR control but using e-values. The main benefit is this procedure allows for arbitrary dependence between the e-values, which is not the case for the BH procedure and p-values.

We are in a multiple testing setup with $k$ hypotheses, each of which has an associated e-value. The procedure rejects the hypotheses with largest $k^{\ast }$ e-values where

$$k^{\ast }=\max \{k:\frac{k{e}_{[k]}}{K}\ge \frac{1}{\alpha }\}$$

and $e_{[1]}\ge e_{[2]}\ge \cdots \ge e_{[K]}$.

This has $\mathbb{E}[F_D/|D|]]\le \alpha K_0/K$, i.e., it has no dependence on ${\ell }_K$, even without the PRDS assumption (this is an improvement over the BH procedure).

The proof is actually quite straightforward: We have $k^{\ast }{e}_{[k^{\ast }]}/K\ge \frac{1}{\alpha }$ by definition. Therefore,

$$\frac{1}{\alpha }\le \frac{k^{\ast }{e}_{k^{\ast }}}{K}\le \frac{k^{\ast }{e}_{\ell }}{K}$$

for all $\ell \le k^{\ast }$. Those indices, $\ell \le k^{\ast }$ are precisely the indices of the rejected hypothesis - the discoveries. That is, $|D|=k^{\ast }$. Therefore, $|D|\ge K/(\alpha e_{[\ell ]})$ for all $\ell \in N$, the set of rejected hypotheses. So,

$$\frac{F_D}{|D|}=\sum _k\frac{1(k\in D\cap N)}{|D|}=\sum _{k\in N}\frac{1(k\in D)}{|D|}\le \sum _{k\in N}\frac{1(k\in D)\alpha E_{[k]}}{K}\le \alpha \sum _{k\in N}\frac{E_{[k]}}{|D|}.$$

Since $\mathbb{E}[E_{[k]}]\le 1$, we have $\mathbb{E}[F_D/|D|]\le \alpha |N|/|D|=\alpha K_0/|D|$.

Boosting e-values If other information is available, we can also boost the e-values. This might be info on the joint distribution, some structure of the problem, etc. But without such additional information, it is better to just use the non-boosted approach above.

References

• False discovery rate control with e-values by Ruodo Wang and Aaditya Ramdas