Two new omnibus tests of uniformity for data on the hypersphere are proposed. The new test statistics exploit closed-form expressions for orthogonal polynomials, feature tuning parameters, and are related to a ``smooth maximum'' function and the Poisson kernel. We obtain exact moments of the test statistics under uniformity and rotationally symmetric alternatives, and give their null asymptotic distributions. We consider approximate oracle tuning parameters that maximize the power of the tests against known generic alternatives and provide tests that estimate oracle parameters through cross-validated procedures while maintaining the significance level. Numerical experiments explore the effectiveness of null asymptotic distributions and the accuracy of inexpensive approximations of exact null distributions. A simulation study compares the powers of the new tests with other tests of the Sobolev class, showing the benefits of the former. The proposed tests are applied to the study of the (seemingly uniform) nursing times of wild polar bears.

翻译：本文提出了两种新的全局检验方法，用于验证超球面上数据的均匀性。新检验统计量利用了正交多项式的闭式表达式，具有调优参数，并与“平滑最大值”函数和泊松核相关。我们在均匀性和旋转对称备择假设下获得了检验统计量的精确矩，并给出了其零渐近分布。我们考虑了近似最优调优参数，这些参数能在已知通用备择假设下最大化检验功效，并通过交叉验证程序估计最优参数，同时维持显著性水平。数值实验验证了零渐近分布的有效性以及精确零分布低成本近似的准确性。模拟研究将新检验的功效与索博列夫类的其他检验进行了比较，展示了前者的优势。提出的检验方法被应用于野生北极熊（看似均匀的）哺育时间研究。