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：本文提出了两种针对超球面数据均匀性的新综合检验方法。新检验统计量利用正交多项式的闭式表达式，特征调优参数，并与“平滑最大值”函数及泊松核相关联。我们推导了均匀分布及旋转对称备择假设下检验统计量的精确矩，并给出了其零渐近分布。针对已知通用备择假设，我们考虑了能够最大化检验功效的近似理想调优参数，并通过交叉验证程序在保持显著性水平的同时提供了估计理想参数的检验方法。数值实验评估了零渐近分布的有效性以及精确零分布廉价近似的准确性。模拟研究比较了新检验方法与索博列夫类其他检验方法的功效，显示了前者的优势。所提出的检验方法被应用于野生北极熊（看似均匀的）哺乳时间研究。