AI 中文总结
研究斯蒂费尔流形上分布的拟合优度检验，基于经验与理论特征函数平方差积分构建框架，导出费希尔 - 宾汉姆分布族检验统计量，给出特殊情况简化式，建立检验程序，模拟验证其准确性与功效，应用于彗星轨道数据。
AI 中文摘要
本文开发了一个用于斯蒂费尔流形上分布的拟合优度检验的综合框架。该方法基于经验特征函数与理论特征函数平方差的积分，产生对所有固定备择假设一致的检验统计量。对于费希尔 - 宾汉姆分布族，导出了检验统计量的显式可计算形式。给出了重要特殊情况（包括矩阵费希尔、矩阵宾汉姆和均匀分布）的简化表达式。在超球面上检验均匀性时，得到了检验统计量的完整渐近分布，实现了计算高效渐近检验。对于一般费希尔 - 宾汉姆分布，为简单和复合假设建立了理论上合理的蒙特卡罗检验程序。模拟研究表明在广泛备择假设下具有准确的第一类错误控制和强大功效。通过对彗星轨道数据的应用说明了所提出方法的实际相关性。
英文摘要
In this article, a comprehensive framework for goodness-of-fit testing for distributions on Stiefel manifolds is developed. The approach is based on integrals of the squared differences between empirical and theoretical characteristic functions, yielding test statistics that are consistent against all fixed alternatives. For the Fisher-Bingham family of distributions, explicit computable forms of the test statistic are derived. Simplified expressions for important special cases, including the matrix Fisher, matrix Bingham, and uniform distributions are provided. In the case of testing uniformity on hyperspheres, we obtain the complete asymptotic distribution of the test statistic, enabling computationally efficient asymptotic testing. For general Fisher-Bingham distributions, we establish theoretically justified Monte Carlo testing procedures for both simple and composite hypotheses. Simulation studies demonstrate accurate Type I error control and strong power across a wide range of alternatives. The practical relevance of the proposed methodology is illustrated by an application to data on the orbits of comets.
Comments25 pages, 1 figure, 1 supplement with proofs
