This work investigates multiple testing from the point of view of 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 properly tuned parameter, and an empirical Bayes $\ell$-value ('local FDR') procedure. We prove that the FDR and FNR have non-symmetric contributions to the testing risk for most procedures, the FNR part being dominant at the boundary. The optimal multiple testing boundary is then investigated for classes of arbitrary sparse signals. A number of extensions, including results for classification losses, are also discussed.

翻译：这项工作从稀有序列模型中小型最大分解率的角度对多重测试进行调查,当测试风险被测量为FDR+FNR总和(False发现率和假负率)时。首先使用流行的β-毫分分分离条件,将所有非零信号至少分解为0美元,我们确定微小最大测试风险的偶然性,从而明确描述从“消失风险的可行多重测试”向“可能的多重测试”的过渡。提供了达到相应最佳边界的适应性多重测试程序:Benjani-Hochberg程序,具有适当调控参数,以及实证性Bayes $@ell$-value (“当地FDR”)程序。我们证明,FDR和FNR对大多数程序的测试风险具有非对称性贡献,FNR部分在边界处于主导地位。然后对任意稀有信号的类别进行最佳多测试边界调查。还讨论了一系列扩展,包括分类损失的结果。