AI 中文总结
研究相关双侧高斯检验中Benjamini-Hochberg程序的最坏情况错误发现率，证明其不存在通用乘法FDR界，构造模型使膨胀因子发散，又给出一类高斯模型的匹配阶上界，表明下界精确。
AI 中文摘要
我们研究了在相关矩阵无其他限制的情况下，将Benjamini-Hochberg程序应用于双侧高斯p值时的最坏情况错误发现率。Dobriban [2026]表明BH并不总是将FDR控制在其名义水平。一个类似的民间猜想是BH能将FDR控制在一个通用乘法常数以内。我们证明了这个猜想是错误的。特别地，我们构造了高斯模型，当q趋于0时，膨胀因子FDR(BH_q)/q会发散。更准确地说，对于所有足够小的q，在假设数量、均值向量和相关矩阵上的上确界至少为cq√log(1/q)，其中c>0为通用常数。最后，对于一类具有任意均值和载荷的广泛的公共因子高斯模型，我们证明了匹配阶的上界FDR(BH_q)=O(q√log(1/q))，因此对于这类模型，下界是精确的。
英文摘要
We study the worst-case false discovery rate (FDR) of the Benjamini-Hochberg procedure for both one- and two-sided Gaussian tests when the correlation matrix is otherwise unrestricted. In each setting we construct a $q$-indexed family of finite Gaussian models whose FDR divided by $q$ diverges as $q\downarrow0$, disproving any universal multiplicative FDR bound. For two-sided tests, the supremum over the number of hypotheses, mean vector, and correlation matrix is at least an explicit $\ell_{=}(q)>q$ satisfying \[ \ell_{=}(q)=\frac{q\sqrt{\log(1/q)}}{2\sqrtπ}+c_\ell q+o(q), \qquad c_\ell=0.6492828\ldots. \] For the one-sided hypotheses $H_i:θ_i\leq0$, a sign-reversed one-common-factor construction gives the stronger explicit lower bound $\ell_{\le}(q)>q$, with \[ \ell_{\le}(q)=\frac{q\sqrt{\log(1/q)}}{\sqrtπ} +\frac q2+o(q). \] Finally, we prove an $O\{q\sqrt{\log(1/q)}\}$ upper bound for the two-sided {one-common-factor} class and the matching upper bound $q\sqrt{\log(1/q)}/\sqrtπ+O(q)$ for the one-sided one-common-factor class.
Comments70 pages, changed the paper title, added lower and upper bounds for one-sided Gaussian tests
