We propose new methods to obtain simultaneous false discovery proportion bounds for knockoff-based approaches. We first investigate an approach based on Janson and Su's $k$-familywise error rate control method and interpolation. We then generalize it by considering a collection of $k$ values, and show that the bound of Katsevich and Ramdas is a special case of this method and can be uniformly improved. Next, we further generalize the method by using closed testing with a multi-weighted-sum local test statistic. This allows us to obtain a further uniform improvement and other generalizations over previous methods. We also develop an efficient shortcut for its implementation. We compare the performance of our proposed methods in simulations and apply them to a data set from the UK Biobank.

翻译：我们提出了新方法，用于在基于knockoffs的方法中获得同时错误发现比例界。首先，我们研究了基于Janson和Su的$k$-族错误率控制方法与插值的方法。随后，通过考虑$k$值的集合对其进行推广，并证明Katsevich和Ramdas的界是该方法的特例，且能够得到一致改进。接着，我们通过使用多加权和局部检验统计量的闭合检验进一步推广了该方法。这使得我们能够在先前方法的基础上获得进一步的一致改进及其他推广。我们还开发了一种高效的简化实现算法。通过模拟实验比较了所提方法的性能，并将其应用于英国生物银行的数据集。