AI 中文总结
研究单位超球面上球形柯西模型的拟合优度检验，利用莫比乌斯变换不变性，通过样本莫比乌斯均值估计参数并变换观测值，应用基于投影的均匀性统计量，推导检验统计量渐近分布，通过实验验证方法有效性。
AI 中文摘要
我们引入了一类针对单位超球面上球形柯西模型的拟合优度检验。所提出的方法利用了球形柯西族在莫比乌斯变换下的不变性：通过样本莫比乌斯均值估计参数后，将观测值进行变换以近似球形均匀性，然后将基于投影的均匀性统计量应用于所得样本。我们表明，在温和条件下，所得检验在原假设下是完全无分布的，因此精确临界值可通过蒙特卡罗模拟任意好地近似。我们研究了莫比乌斯均值作为总体泛函，在温和条件下确定其存在性和唯一性，并证明其经验对应物的等变性和渐近线性，它与球形柯西最大似然估计器一致。我们推导了检验统计量的渐近原分布，并表明在去除对应于球形柯西模型切空间的一次球谐分量后，它与基础均匀性统计量的渐近原分布一致。我们建立了针对固定备择假设的一致性，并通过相邻备择假设的球谐分解来刻画局部功效。蒙特卡罗实验证明了渐近近似的有限样本准确性和所提出检验的经验功效。还处理了一个实际数据示例。
英文摘要
We introduce a class of goodness-of-fit tests for the spherical Cauchy model on the unit hypersphere. The proposed procedures exploit the invariance of the spherical Cauchy family under Möbius transformations: after estimating the parameter by the sample Möbius mean, the observations are transformed to approximate spherical uniformity, and a projection-based uniformity statistic is applied to the resulting sample. We show that, under mild conditions, the resulting tests are exactly distribution-free under the null hypothesis, so that exact critical values can be arbitrarily well approximated by Monte Carlo simulation. We study the Möbius mean as a population functional, establish its existence and uniqueness under mild conditions, and prove equivariance and asymptotic linearity of its empirical counterpart, which coincides with the spherical Cauchy maximum likelihood estimator. We derive the asymptotic null distribution of the test statistic and show that it coincides with that of the underlying uniformity statistic after removing the degree-one spherical-harmonic component, which corresponds to the tangent space of the spherical Cauchy model. We establish consistency against fixed alternatives and characterize local powers through the spherical-harmonic decomposition of contiguous alternatives. Monte Carlo experiments demonstrate the finite-sample accuracy of the asymptotic approximations and the empirical power of the proposed tests. A real data example is treated.
Comments44 pages (a main paper of 21 pages and several appendices)
