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.

翻译：本文基于Le Cam方法，构建了高维环面上的对称性最优检验。我们同时处理了对称中心已知与未知两种情况。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同样适用。我们研究了所提检验在原假设与局部备择假设下的渐近行为，并利用Stein方法推导了检验统计量的精确（未知）分布与其渐近分布之间分布距离的定量界。通过模拟研究评估了检验的有限样本性能，并通过对蛋白质折叠数据的应用展示了其实用价值。此外，我们建立了一个关于函数二次均值可微性的普适性结果，该性质是应用Le Cam方法的关键理论基础。