Identifying the most powerful test in multiple hypothesis testing under strong family-wise error rate (FWER) control is a fundamental problem in statistical methodology. State-of-the-art approaches formulate this as a constrained optimisation problem, for which a dual problem with strong duality has been established in a general sense. However, a constructive method for solving the dual problem is lacking, leaving a significant computational gap. This paper fills this gap by deriving novel, necessary optimality conditions for the dual optimisation. We show that these conditions motivate an efficient coordinate-wise algorithm for computing the optimal dual solution, which, in turn, provides the most powerful test for the primal problem. We prove the linear convergence of our algorithm, i.e., the computational complexity of our proposed algorithm is proportional to the logarithm of the reciprocal of the target error. To the best of our knowledge, this is the first time such a fast and computationally efficient algorithm has been proposed for finding the most powerful test with family-wise error rate control. The method's superior power is demonstrated through simulation studies, and its practical utility is shown by identifying new, significant findings in both clinical and financial data applications.

翻译：在强族错误率控制下识别多重假设检验中的最强检验是统计方法论中的一个基本问题。现有前沿方法将此问题表述为一个约束优化问题，其一般意义上已建立具有强对偶性的对偶问题。然而，目前缺乏求解该对偶问题的构造性方法，导致显著的计算空白。本文通过推导对偶优化的新颖必要最优性条件填补了这一空白。我们证明这些条件可激发一种高效的坐标式算法，用于计算最优对偶解，进而为原始问题提供最强检验。我们证明了该算法的线性收敛性，即所提算法的计算复杂度与目标误差倒数的对数成正比。据我们所知，这是首次提出如此快速且计算高效的算法来寻找具有族错误率控制的最强检验。通过模拟研究验证了该方法在检验功效上的优越性，并在临床与金融数据应用中识别出新的显著发现，展现了其实用价值。