Abstract A test of modality of rotationally symmetric distributions on hyperspheres is proposed. The test is based on a modified multivariate kurtosis defined for directional data on S d . We first reveal a relationship between the multivariate kurtosis and the types of modality for Euclidean data. In particular, the kurtosis of a rotationally symmetric distribution with decreasing sectional density is greater than the kurtosis of the uniform distribution, while the kurtosis of that with increasing sectional density is less. For directional data, we show an asymptotic normality of the modified spherical kurtosis, based on which a large-sample test is proposed. The proposed test of modality is applied to the problem of preventing overfitting in non-geodesic dimension reduction of directional data. The proposed test is superior than existing options in terms of computation times, accuracy and preventing overfitting. This is highlighted by a simulation study and two real data examples.

中文翻译：

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超球面上旋转对称分布模态的峰度检验

摘要 提出了一种检验超球面上旋转对称分布模态的方法。该检验基于为 S d 上的方向数据定义的修正多变量峰度。我们首先揭示了欧几里得数据的多变量峰度和模态类型之间的关系。特别是旋转对称分布的峰度随着截面密度的减小而减小，其峰度大于均匀分布的峰度，而随着截面密度增大，其峰度小于均匀分布的峰度。对于定向数据，我们展示了修正球面峰度的渐近正态性，并在此基础上提出了大样本检验。提出的模态测试应用于防止方向数据非测地线降维中的过度拟合问题。建议的测试在计算时间、准确性和防止过度拟合方面优于现有选项。模拟研究和两个真实数据示例突出了这一点。