标题： 在环面上传统对称性的最优检验构造及其定量误差界限 显示英文标题

标题： Construction of optimal tests for symmetry on the torus and their quantitative error bounds

摘要： 在本文中，我们开发了在高维环面上对称性的最优检验，利用了Le Cam的方法论。我们考虑了对称中心已知和未知的两种情况。这些检验不仅在给定的参数假设下有效，而且在非常广泛的对称分布类中也有效。我们研究了所提出的检验在原假设和局部替代假设下的渐近行为，并通过Stein方法推导了检验统计量的精确（未知）分布与其渐近对应物之间的分布距离的定量界限。通过模拟研究评估了检验的有限样本性能，并通过应用于蛋白质折叠数据展示了其实际用途。此外，我们建立了一个广泛适用的结果，关于函数的二次均值可微性，这是Le Cam方法使用的关键性质。 显示英文摘要

摘要： In this paper, we develop optimal tests for symmetry on the hyper-dimensional torus, leveraging Le Cam's methodology. We address both scenarios where the center of symmetry is known and where it is unknown. These tests are not only valid under a given parametric hypothesis but also under a very broad class of symmetric distributions. The asymptotic behavior of the proposed tests is studied both under the null hypothesis and local alternatives, and we derive quantitative bounds on the distributional distance between the exact (unknown) distribution of the test statistic and its asymptotic counterpart using Stein's method. The finite-sample performance of the tests is evaluated through simulation studies, and their practical utility is demonstrated via an application to protein folding data. Additionally, we establish a broadly applicable result on the quadratic mean differentiability of functions, a key property underpinning the use of Le Cam's approach.