关于超球面上均匀性的Stein检验
On Stein's test of uniformity on the hypersphere
AI总结：
提出基于Laplace-Beltrami算子的Stein特征检验超球面均匀性，利用Gegenbauer多项式调和分解，导出检验统计量的渐近分布，并通过调节参数增强对多种备择假设的检验功效。
摘要：
我们基于与Laplace-Beltrami算子相关的Stein特征，提出了一种新的超球面均匀性检验。我们为此特征确定了一个与矩生成函数相关的充分检验函数类。利用该算子的特征函数获得关于Gegenbauer多项式的调和分解，我们证明了所提出的方法属于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渐近分布的闭式级数表示。为了增强对一系列备择假设的检验功效，我们在特征中引入了一个调节参数，并研究其对拒绝概率的影响。我们讨论了数据驱动的参数选择策略，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进行比较。额外的数值实验将所提出的检验与竞争性的Sobolev类程序进行了比较，突出了其具有明显优势的场景。
英文摘要：
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 series representations for the asymptotic 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.
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