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：本文提出了两种适用于超球面数据均匀性检验的新综合检验方法。新检验统计量利用正交多项式的闭式表达式、特征调节参数，并与"平滑最大值"函数及泊松核相关联。我们推导了在均匀性和旋转对称备择假设下检验统计量的精确矩，并给出了零假设下的渐近分布。通过考虑近似最优调节参数（针对已知通用备择假设可最大化检验功效），我们提供了在维持显著性水平的同时通过交叉验证程序估计最优参数的检验方法。数值实验验证了零假设渐近分布的有效性及精确零假设分布廉价近似的准确性。模拟研究比较了新检验与索伯列夫类其他检验的功效，结果表明前者具有显著优势。所提方法被应用于野生北极熊（看似均匀分布的）哺乳时间研究。