AI 中文总结
本文针对任意依赖下双侧高斯均值检验，推导了BH程序FDR的依赖自适应界，提出了协方差未知时具有限样本FDR保证的置信界移位BH程序，验证了其性能优势。
AI 中文摘要
Sarkar和Zhang（2025）的最新研究引入了原假设下的正尾部依赖（PTDN），并针对已知协方差结构下的双侧高斯z检验和t检验，开发了广义移位Benjamini-Hochberg（BH）程序。本文进一步推导该框架的相关结论：首先，我们针对原始BH程序的错误发现率（FDR），基于条件方差参数τ_i=1-R_i²（其中R_i²是第i个统计量与其余坐标之间的多重相关系数平方），推导了显式的依赖自适应上下界。这些界在独立情形下可还原为精确的BH FDR，且提供了有限样本、协方差特定的信息，可补充通用界；同时，我们确定了移位BH的坐标特定校准能较原始BH程序获得拒绝优势的条件。其次，我们考虑实际中重要的协方差矩阵未知但存在独立Wishart估计量的情形，利用τ_i的联合下置信界，构造了置信界移位BH程序，并建立了有限样本FDR控制。据我们所知，这是首个针对完全未知且独立估计的协方差矩阵下的双侧高斯均值检验，具有有限样本保证的移位BH类程序。数值研究验证了协方差自适应界的行为、移位BH较BH的潜在优势，以及未知协方差下置信界移位的性能。
英文摘要
The recent work of Sarkar and Zhang (2025) introduced Positive Tail Dependence Under the Null (PTDN) and developed Generalized Shifted Benjamini-Hochberg (BH) procedures for two-sided Gaussian $z$- and $t$-testing under known covariance structures. This paper develops further consequences of that framework. First, we derive explicit dependence-adaptive lower and upper bounds for the FDR of the original BH procedure in terms of the conditional variance parameters $τ_i=1-R_i^2$, where $R_i^2$ is the squared multiple correlation between the $i$th statistic and the remaining coordinates. These bounds recover the exact BH FDR under independence and provide finite-sample, covariance-specific information complementary to generic bounds. We also identify conditions under which the coordinate-specific calibration of shifted BH can provide a rejection advantage over the original BH procedure. Second, we consider the practically important setting in which the covariance matrix is unknown but an independent Wishart estimator is available. Using simultaneous lower confidence bounds for the $τ_i$'s, we construct a confidence-bound shifted BH procedure and establish finite-sample FDR control. To our knowledge, this is the first shifted-BH-type procedure with a finite-sample guarantee for two-sided Gaussian mean testing under a completely unknown covariance matrix estimated independently. Numerical studies illustrate the behavior of the covariance-adaptive bounds, the potential advantage of shifted BH over BH, and the performance of confidence-bound shifting under unknown covariance.
Comments18 pages, 4 figures
