False discovery rate control with unknown null distribution: is it possible to mimic the oracle?

Abstract: Classical multiple testing theory prescribes the null distribution, which is often a too stringent assumption for nowadays large scale experiments. This paper presents theoretical foundations to understand the limitations caused by ignoring the null distribution, and how it can be properly learned from the (same) data-set, when possible. We explore this issue in the case where the null distributions are Gaussian with an unknown rescaling parameters (mean and variance) and the alternative distribution is let arbitrary. While an oracle procedure in that case is the Benjamini Hochberg procedure applied with the true (unknown) null distribution, we pursue the aim of building a procedure that asymptotically mimics the performance of the oracle (AMO in short). Our main result states that an AMO procedure exists if and only if the sparsity parameter $k$ (number of false nulls) is of order less than $n/\log(n)$, where $n$ is the total number of tests. Further sparsity boundaries are derived for general location models where the shape of the null distribution is not necessarily Gaussian. Given our impossibility results, we also pursue a weaker objective, which is to find a confidence region for the oracle. To this end, we develop a distribution-dependent confidence region for the null distribution. As practical by-products, this provides a goodness of fit test for the null distribution, as well as a visual method assessing the reliability of empirical null multiple testing methods. Our results are illustrated with numerical experiments and a companion vignette \cite{RVvignette2020}.