超球面上均匀性检验的一种通用极大投影方法
A general maximal projection approach to uniformity testing on the hypersphere
摘要：
提出了一种基于最大投影的$d$维单位超球$\mathcal{S}^{d-1}$均匀性检验方法.这种方法给出了一个统一的观点，经典的均匀性测试的瑞利和宾汉，它链接到措施的多元偏度和峰度。利用巴拿赫空间值随机过程的极限定理，在原假设下导出了极限分布，并提出了利用球谐函数理论模拟极限过程的策略.我们研究的行为下连续和固定的替代品，并显示的一致性的测试程序的某些类别的替代品。第一次在球的均匀性测试，我们得到当地巴哈杜尔效率报表。我们评估的理论研究结果和经验的权力的程序在一个广泛的竞争蒙特卡罗模拟研究，最后，应用新的测试的数据集上的中点的月球上的大陨石坑。
英文摘要：
We propose a novel approach to uniformity testing on the $d$-dimensional unit hypersphere $\mathcal{S}^{d-1}$ based on maximal projections. This approach gives a unifying view on the classical uniformity tests of Rayleigh and Bingham, and it links to measures of multivariate skewness and kurtosis. We derive the limiting distribution under the null hypothesis using limit theorems for Banach space valued stochastic processes and we present strategies to simulate the limiting processes by applying results on the theory of spherical harmonics. We examine the behavior under contiguous and fixed alternatives and show the consistency of the testing procedure for some classes of alternatives. For the first time in uniformity testing on the sphere, we derive local Bahadur efficiency statements. We evaluate the theoretical findings and empirical powers of the procedures in a broad competitive Monte Carlo simulation study and, finally, apply the new tests to a data set on midpoints of large craters on the moon.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
