Many modern applications require using data to select the statistical tasks and make valid inference after selection. In this article, we provide a unifying approach to control for a class of selective risks. Our method is motivated by a reformulation of the celebrated Benjamini-Hochberg (BH) procedure for multiple hypothesis testing as the fixed point iteration of the Benjamini-Yekutieli (BY) procedure for constructing post-selection confidence intervals. Building on this observation, we propose a constructive approach to control extra-selection risk (where selection is made after decision) by iterating decision strategies that control the post-selection risk (where decision is made after selection). We show that many previous methods and results are special cases of this general framework, and we further extend this approach to problems with multiple selective risks. Our development leads to two surprising results about the BH procedure: (1) in the context of one-sided location testing, the BH procedure not only controls the false discovery rate at the null but also at other locations for free; (2) in the context of permutation tests, the BH procedure with exact permutation p-values can be well approximated by a procedure which only requires a total number of permutations that is almost linear in the total number of hypotheses.

翻译：许多现代应用需要利用数据选择统计任务，并在选择后做出有效推断。本文提出了一种统一方法来控制一类选择性风险。我们的方法源于对著名的Benjamini-Hochberg（BH）多重假设检验程序的重构，将其表述为构建选择后置信区间的Benjamini-Yekutieli（BY）程序的不动点迭代。基于这一观察，我们提出了一种构建性方法：通过迭代控制选择后风险（即决策在选择之后进行）的决策策略，来控制额外选择风险（即选择在决策之后进行）。我们证明许多现有方法和结果都是这一通用框架的特例，并进一步将该方法推广到具有多重选择性风险的问题。我们的研究得出了关于BH程序的两个意外结果：（1）在单侧位置检验中，BH程序不仅能在零假设处控制错误发现率，还能免费在其他位置实现控制；（2）在置换检验中，使用精确置换p值的BH程序可以通过一个仅需总置换次数几乎与假设总数呈线性关系的程序来良好近似。