MaxT is a highly popular resampling-based multiple testing procedure, which controls the Familywise Error Rate (FWER) and is powerful under dependence. This paper generalizes maxT to what we term ``multi-resolution'' False Discovery eXceedance (FDX) control. Basic FDX control means ensuring that the FDP -- the proportion of false discoveries among all rejections -- is at most $γ$ with probability at least $1-α$. Here $γ$ and $α$ are prespecified, small values between 0 and 1. The proposed method is in addition simultaneous, in the following way: the procedure outputs a single rejection threshold $q$, but ensures that with probability $1-α$, simultaneously over all stricter thresholds, the corresponding FDPs are also below $γ$. In particular, for a small set of hypotheses, the FDP bound is 0, i.e., the FWER is 0. Despite these additional, simultaneous guarantees, our method has power comparable to Romano-Wolf, the most powerful non-simultaneous FDX method. Further, our method is valid under the same assumptions. Thus, this paper shows that FDX methods can often be made simultaneous almost for free. The proposed method can be formulated as an extension of simultaneous approaches such as Hemerik, Solari and Goeman (2019), for the first time allowing for confidence envelopes with a data-dependent shape -- thus resolving a major limitation of such methods.

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