A nonparametric test for elliptical distribution based on kernel embedding of probabilities
基于概率核嵌入的椭圆分布非参数检验AI 解读 ↓
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基于概率核嵌入的椭圆分布非参数检验
- 研究问题
- 研究如何一般性地检验多变量数据是否服从椭圆分布，并在维度随样本量增长时保持有效的检验性质。
- 方法与思路
- 作者用概率核嵌入刻画标准化后方向与长度的独立性及方向的球面均匀性，并以 von Mises 展开推导渐近分布。
- 材料陈述的贡献
- 摘要称建立拒绝域构造、检验有效性与一致性，并给出支持增长维度分析的集中界，开展模拟及 SENIC 数据应用。
- 阅读关注点
- 阅读全文应核对中心与尺度估计、核函数条件、拒绝域校准，以及高维情形允许的维度增长和备择条件。
- 阅读边界
- 仅据 OpenAlex 索引摘要，稿件版本未注明；摘要未给出维度增长率、核选择或检验比较数值，无法判断普遍性能优势。
原文摘要
Elliptical distribution is a basic assumption underlying many multivariate statistical methods. For example, in sufficient dimension reduction and statistical graphical models, this assumption is routinely imposed to simplify the data dependence structure. Before applying such methods, we need to decide whether the data are elliptically distributed. Currently existing tests either focus exclusively on spherical distributions, or rely on bootstrap to determine the null distribution, or require specific forms of the alternative distribution. In this paper, we introduce a general nonparametric test for elliptical distribution based on kernel embedding of the probability measure that embodies the two properties that characterize an elliptical distribution: namely, after centering and rescaling, (1) the direction and length of the random vector are independent, and (2) the directional vector is uniformly distributed on the unit sphere. We derive the asymptotic distributions of the test statistic via von Mises expansion, develop the sample-level procedure to determine the rejection region, and establish the consistency and validity of the proposed test. We also develop the concentration bounds of the test statistic, allowing the dimension to grow with the sample size, and further establish the consistency in this high-dimension setting. We compare our method with several existing methods via simulation studies, and apply our test to a SENIC dataset with and without a transformation aimed to achieve ellipticity.