# Inference while controlling False Discovery rates given known dependence between test statistics

I have a number $n$ of standard normal test statistics $\boldsymbol{\beta}$, each of which belonging to a hypothesis I want to test. So under H$_0$

$$\boldsymbol{\beta} \sim N(\mathbf{0}, \boldsymbol{\Sigma})$$

where $\boldsymbol{\Sigma}$ has been estimated using maximum likelihood theory. How can I test all $n$ hypotheses separately, while controlling the false discovery rate but maintaining maximal power by leveraging my knowledge of $\boldsymbol{\Sigma}$?

I've been looking through the literature but can't quite seem to find what I want.

What I do not want is:

• A chi-squared omnibus test
• A very general correction on the p-values that is too conservative
• A procedure that assumes $\boldsymbol{\Sigma}$ unknown
• A procedure based on dependence in the data
• Are you conflating conducting a group of pairwise tests of the same kind with control of the FDR? The former provides p-values – including for dependent data (e.g., paired t test) – for each pairwise test, while the latter is a procedure for controlling false positives due to having more than a single test, regardless of whether the specific tests were dependent or not. – Alexis May 16 '18 at 16:19
• The Benjamini-Hochberg FDR adjustment method only cares if the tests themselves have negatively dependent rejection probabilities; a condition which is difficult to imagine even hypothtically. (Especially note amoeba's answer, and ignore Chris C's which is more or less irrelevant and contains misinformation). – Alexis May 16 '18 at 16:24
• @Alexis No I am not conflating both, I really want FDR control for correlated tests. I know Benjamini-Hochberg or if necessary Benjamini-Yekutieli will work, but I am concerned about power. My feeling is that there must be something more powerful given that you know the full covariance structure. – Knarpie May 16 '18 at 19:07