Whiteout：fixed-X仿冒品什么时候会失败？
Whiteout: when do fixed-X knockoffs fail?
摘要：
山寨方法的核心优势是其几乎无限的可定制性，允许分析师利用机器学习算法和领域知识，而不会威胁该方法的鲁棒有限样本错误发现率控制保证。虽然以前的几个作品已经调查了制度，其中特定的实施方案的仿冒品是可证明的强大，一般的负面结果是更难以获得这样一个灵活的方法。在这项工作中，我们重铸固定的$X$敲除过滤器的高斯线性模型作为一个有条件的后选择推理方法。它将用户生成的高斯噪声添加到普通最小二乘估计器$\hatβ$以获得具有不相关条目的“白化”估计器$\widetildeβ$，并使用$\text{sgn}(\widetildeβ_j)$作为$H_j:\; β_j = 0$的检验统计量来执行推断。我们证明了我们的美白配方和更标准的配方，涉及负控制预测变量之间的等价性，显示如何固定的$X$敲框架可以用于多个测试的任何问题（渐近）多变量高斯参数估计。依靠这个角度来看，我们得到的第一个负面的结果，普遍上限的权力，所有固定的$X$仿制品的方法，而不考虑的选择由分析师。我们的研究结果大致表明，如果$\text{Var}(\hatβ)$的前导特征值很大，前导特征向量很密集，那么就没有办法在不不可挽回地擦除几乎所有信号的情况下消除$\hatβ$，从而使$\text{sgn}(\widetildeβ_j)$信息量太少，无法进行准确的推断。我们给出的条件下，任何固定的$X$敲除方法的真阳性率（TPR）必须收敛到零，即使TPR的Bonferroni校正的多重测试趋于一，我们探讨了几个例子说明这种现象。
英文摘要：
A core strength of knockoff methods is their virtually limitless customizability, allowing an analyst to exploit machine learning algorithms and domain knowledge without threatening the method's robust finite-sample false discovery rate control guarantee. While several previous works have investigated regimes where specific implementations of knockoffs are provably powerful, general negative results are more difficult to obtain for such a flexible method. In this work we recast the fixed-$X$ knockoff filter for the Gaussian linear model as a conditional post-selection inference method. It adds user-generated Gaussian noise to the ordinary least squares estimator $\hatβ$ to obtain a "whitened" estimator $\widetildeβ$ with uncorrelated entries, and performs inference using $\text{sgn}(\widetildeβ_j)$ as the test statistic for $H_j:\; β_j = 0$. We prove equivalence between our whitening formulation and the more standard formulation involving negative control predictor variables, showing how the fixed-$X$ knockoffs framework can be used for multiple testing on any problem with (asymptotically) multivariate Gaussian parameter estimates. Relying on this perspective, we obtain the first negative results that universally upper-bound the power of all fixed-$X$ knockoff methods, without regard to choices made by the analyst. Our results show roughly that, if the leading eigenvalues of $\text{Var}(\hatβ)$ are large with dense leading eigenvectors, then there is no way to whiten $\hatβ$ without irreparably erasing nearly all of the signal, rendering $\text{sgn}(\widetildeβ_j)$ too uninformative for accurate inference. We give conditions under which the true positive rate (TPR) for any fixed-$X$ knockoff method must converge to zero even while the TPR of Bonferroni-corrected multiple testing tends to one, and we explore several examples illustrating this phenomenon.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
