  An important limitation of standard multiple testing procedures is that the
null distribution should be known. Here, we consider a
  null distribution-free approach for multiple testing in the following
semi-supervised setting: the user does not know the null distribution, but has
at hand a sample drawn from this null distribution. In practical situations,
this null training sample (NTS) can come from previous experiments, from a part
of the data under test, from specific simulations, or from a sampling process.
In this work, we present theoretical results that handle such a framework, with
a focus on the false discovery rate (FDR) control and the Benjamini-Hochberg
(BH) procedure. First, we provide upper and lower bounds for the FDR of the BH
procedure based on empirical $p$-values. These bounds match when $\alpha
(n+1)/m$ is an integer, where $n$ is the NTS sample size and $m$ is the number
of tests. Second, we give a power analysis for that procedure suggesting that
the price to pay for ignoring the null distribution is low when $n$ is
sufficiently large in front of $m$; namely $n\gtrsim m/(\max(1,k))$, where $k$
denotes the number of ``detectable'' alternatives. Third, to complete the
picture, we also present a negative result that evidences an intrinsic
transition phase to the general semi-supervised multiple testing problem {and
shows that the empirical BH method is optimal in the sense that its performance
boundary follows this transition phase}. Our theoretical properties are
supported by numerical experiments, which also show that the delineated
boundary is of correct order without further tuning any constant. Finally, we
demonstrate that our work provides a theoretical ground for standard practice
in astronomical data analysis, and in particular for the procedure proposed in
\cite{Origin2020} for galaxy detection.
