关于超球面上均匀性的新 omnibus 检验
On new omnibus tests of uniformity on the hypersphere
AI总结：
本文提出两种超球面均匀性新omnibus检验，基于正交多项式闭式等构造，推导相关分布与调优方法，实验验证其功效优势，并应用于野生北极熊哺乳时间研究。
摘要：
本文提出了两种针对超球面上数据的均匀性新omnibus检验。新的检验统计量利用了正交多项式的闭式表达式，具备调优参数，且与“光滑最大值”函数及Poisson核相关。我们推导了检验统计量在均匀分布和旋转对称备择假设下的精确矩，并给出了它们的零假设渐近分布。我们研究了能最大化检验针对已知通用备择假设的功效的近似oracle调优参数，并提供了通过交叉验证程序估计oracle参数、同时维持显著性水平的检验方法。数值实验探究了零假设渐近分布的有效性，以及精确零分布的低成本近似的准确性。模拟研究将新检验的功效与Sobolev类的其他检验进行了对比，体现了新检验的优势。所提出的检验被应用于野生北极熊（看似均匀的）哺乳时间的研究中。
英文摘要：
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.
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