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.

翻译：MaxT是一种流行的基于重抽样的多重检验方法，它能控制族系错误率（FWER）且在相依条件下具有较高功效。本文将由MaxT推广至我们称为"多分辨率"的假阳性超出率（FDX）控制。基本FDX控制指确保在所有拒绝中假阳性比例（FDP）以至少1-α的概率不超过γ，其中γ与α是预先设定的介于0和1之间的较小数值。所提方法还具有如下同步特性：程序输出单一拒绝阈值q，同时以1-α的概率保证所有更严格阈值对应的FDP也均低于γ。特别地，对于小规模假设集合，FDP界限为0（即FWER为0）。尽管增加了这些同步保证，本方法仍具有与最强非同步FDX方法Romano-Wolf相当的功效，且在相同假设下保持有效性。这表明FDX方法通常可以几乎无代价地实现同步性。所提方法可视为Hemerik、Solari和Goeman（2019）等同步方法的拓展，首次允许置信包络具有数据依赖形状——从而解决了该类方法的主要局限。