We propose a new test of uniformity on the hypersphere based on a Stein characterization associated with the Laplace--Beltrami operator. We identify a sufficient class of test functions for this characterization, linked to the moment generating function. Exploiting the operator's eigenfunctions to obtain a harmonic decomposition in terms of Gegenbauer polynomials, we show that the proposed procedure belongs to the class of Sobolev tests. We derive closed-form expressions for the distribution of the test statistic under the null hypothesis and under fixed alternatives. To enhance power against a range of alternatives, we introduce a tuning parameter into the characterization and study its impact on rejection probabilities. We discuss data-driven strategies for selecting this parameter to maximize rejection rates for a given alternative and compare the resulting performance with that of related parametric tests. Additional numerical experiments compare the proposed test with competing Sobolev-class procedures, highlighting settings in which it offers clear advantages.

翻译：本文提出了一种基于与拉普拉斯-贝尔特拉米算子相关的Stein刻画的新型超球面均匀性检验方法。我们为该刻画确定了一类与矩生成函数相关的充分检验函数族。通过利用算子的特征函数获得基于盖根堡多项式的调和分解，我们证明了所提方法属于Sobolev检验类。我们推导了零假设和固定备择假设下检验统计量分布的闭式表达式。为提升对多种备择假设的检验功效，我们在刻画中引入了调节参数并研究其对拒绝概率的影响。我们讨论了通过数据驱动策略选择该参数以最大化给定备择假设下的拒绝率，并将所得性能与相关参数检验方法进行比较。额外的数值实验将所提检验方法与同类Sobolev检验程序进行对比，明确了其具有显著优势的应用场景。