Simultaneously testing $K$ hypotheses while controlling the family-wise error rate is a fundamental problem in statistics. Existing procedures (Bonferroni, Holm, Hochberg, Hommel) provide valid control but sacrifice power, increasingly so as $K$ grows, because they base decisions on marginal $p$-value ranks rather than the joint likelihood. Rosset et al. (2022) formulated the most powerful family-wise-error-rate-controlling test as a dual program and proved the existence of an optimal dual vector $μ^*$, but left its computation as an open problem. We solve this problem for $K$ exchangeable hypotheses. The key insight is that the family-wise error rate constraint coefficients $b_{l,k}(\vec{u})$ admit closed-form expressions through elementary symmetric polynomials of the likelihood-ratio values $g(u_1), \ldots, g(u_K)$. This algebraic structure implies a global monotonicity theorem: the target functions $F_γ(μ) = {\rm FWER}_γ(\vec{D}^μ)$ are simultaneously non-increasing in every component of $μ$, for arbitrary $K$, which guarantees unique coordinate-wise roots and enables a bisection-based coordinate-descent algorithm with $O(\log \varepsilon^{-1})$ convergence rate. The relative power gain over Hommel's method grows from 15\% at $K{=}3$ to 84\% at $K{=}12$. Applications to replication studies, a clinical trial, and a replicability assessment illustrate both the power gains and the role of the exchangeability assumption.

翻译：同时检验$K$个假设并控制家族误差率是统计学中的基本问题。现有方法（如Bonferroni、Holm、Hochberg、Hommel）虽能提供有效控制，但随着$K$增大，因依赖边际$p$值排序而非联合似然函数，其检验效能逐渐降低。Rosset等（2022）将最有效的家族误差率控制检验表述为对偶规划，证明了最优对偶向量$μ^*$的存在性，但未给出具体计算方法。针对$K$个可交换假设，我们解决了该计算问题。关键发现是：家族误差率约束系数$b_{l,k}(\vec{u})$可通过似然比值$g(u_1), \ldots, g(u_K)$的初等对称多项式获得闭式表达式。这一代数结构引出了全局单调性定理：对于任意$K$，目标函数$F_γ(μ) = {\rm FWER}_γ(\vec{D}^μ)$在$μ$的每个分量上均为非递增函数，从而保证了唯一的坐标根存在性，并支持基于二分法的坐标下降算法，收敛速率达$O(\log \varepsilon^{-1})$。与Hommel方法相比，本方法的相对效能增益从$K{=}3$时的15%提升至$K{=}12$时的84%。通过复制性研究、临床试验和可重复性评估的应用案例，我们验证了效能提升效果及可交换性假设的作用。