Generalized Boundary FDR Control under Arbitrary Dependence: An Approach on Closure Principle

False discovery rate (FDR) is a cornerstone of modern multiple testing. However, it often fails to guarantee the reliability of ``marginal" discoveries that lie at the boundary of the rejection set, which are often crucial in high-precision applications. While recent works (Soloff et al., 2024; Xiang et al., 2025) introduced the boundary false discovery rate (bFDR) to control the error probability at the marginal discovery, their method relies on restrictive assumptions such as independence or specific prior distributions. In this paper, we first propose $k$-bFDR, a novel generalization that controls the error probability of the $k$ least significant discoveries. We then provide a systematic investigation into the theoretical relationship between $k$-bFDR and existing error metrics. Furthermore, building upon the closure principle, we develop Domino, a unified framework that guarantees $k$-bFDR control under arbitrary dependence, applicable for both p-values and e-values. We prove the theoretical validity of the proposed Domino algorithm and demonstrate through extensive numerical experiments that it consistently achieves rigorous $k$-bFDR control while identifying trustworthy marginal discoveries. Analyses of real data reveal that $k$-bFDR control yields higher-quality rejection sets with greater practical significance.

Imagine a scientist testing hundreds of potential new drugs at once. Standard statistical tools can tell them how many total mistakes they likely made, but they cannot guarantee whether the very weakest drug on their list is actually effective. That borderline discovery might be just a fluke, yet it is often the most interesting one for further research. We face this same problem with many types of data-driven discoveries. To solve this, we introduce a new method called Domino, which ensures that even the borderline discoveries are likely to be true. Unlike previous methods that fail when data points are related in complex ways, Domino works no matter how tangled the relationships are. Our framework gives researchers confidence that even their most marginal discoveries are likely true. This matters because in fields like medicine or genetics, chasing a false lead wastes time and resources. With Domino, scientists can now build a set of findings where every single discovery, down to the last one, meets a high standard of trust.