  In many applications, the process of identifying a specific feature of
interest often involves testing multiple hypotheses for their joint statistical
significance. Examples include mediation analysis which simultaneously examines
the existence of the exposure-mediator and the mediator-outcome effects, and
replicability analysis aiming to identify simultaneous signals that exhibit
statistical significance across multiple independent experiments. In this
study, we present a new approach called joint mirror (JM) procedure that
effectively detects such features while maintaining false discovery rate (FDR)
control in finite samples. The JM procedure employs an iterative method that
gradually shrinks the rejection region based on progressively revealed
information until a conservative estimate of the false discovery proportion
(FDP) is below the target FDR level. Additionally, we introduce a more
stringent error measure, known as the modified FDR (mFDR), which assigns
weights to each false discovery based on its number of null components. We
demonstrate that, under appropriate assumptions, the JM procedure controls the
mFDR in finite samples. To implement the JM procedure, we propose an efficient
algorithm that can incorporate partial ordering information. Through extensive
simulations, we demonstrate that our procedure effectively controls the mFDR
and enhances statistical power across various scenarios. Finally, we showcase
the utility of our method by applying it to real-world mediation and
replicability analyses.
