This work investigates multiple testing by considering minimax separation rates in the sparse sequence model, when the testing risk is measured as the sum FDR+FNR (False Discovery Rate plus False Negative Rate). First using the popular beta-min separation condition, with all nonzero signals separated from $0$ by at least some amount, we determine the sharp minimax testing risk asymptotically and thereby explicitly describe the transition from "achievable multiple testing with vanishing risk" to "impossible multiple testing". Adaptive multiple testing procedures achieving the corresponding optimal boundary are provided: the Benjamini--Hochberg procedure with a properly tuned level, and an empirical Bayes $\ell$-value (`local FDR') procedure. We prove that the FDR and FNR make non-symmetric contributions to the testing risk for most optimal procedures, the FNR part being dominant at the boundary. The multiple testing hardness is then investigated for classes of arbitrary sparse signals. A number of extensions, including results for classification losses and convergence rates in the case of large signals, are also investigated.

翻译：本文研究稀疏序列模型中的多重检验问题，以检验风险 FDR+FNR（错误发现率加错误否定率）之和为测度，重点关注极小化可分离速率。首先，基于流行的 β-min 分离条件（所有非零信号与 0 的距离至少达到某个阈值），我们渐近地确定了锐利极小化检验风险，从而明确刻画了从“可实现风险趋零的多重检验”到“不可实现多重检验”的转变。本文提供了达到相应最优边界的自适应多重检验程序：适当调节阈值的 Benjamini–Hochberg 程序，以及经验贝叶斯 ℓ 值（局部 FDR）程序。我们证明，对于大多数最优程序，FDR 和 FNR 对检验风险的贡献是非对称的，其中 FNR 部分在边界处占主导地位。随后，针对任意稀疏信号类别研究了多重检验的困难程度。此外，本文还探讨了若干扩展，包括分类损失下的结果以及大信号情形下的收敛速率。