Multivariate Nonparametric Tests of Independence
多变量独立性的非参数检验AI 解读 ↓
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多变量独立性的非参数检验
- 研究问题
- 如何将 Kendall 与 Spearman 的关联统计量扩展到两个随机向量的独立性检验，并构造便于计算的稳健检验方法？
- 方法与思路
- 作者分别利用方向间比例及仿射等变的多变量符号与秩构造统计量，研究渐近分布、Pitman 效率和小样本置换分布。
- 材料陈述的贡献
- 摘要提出新的多变量独立性统计量，称两类方法在椭圆对称条件下似乎渐近等价，并以模拟和实例说明其表现。
- 阅读关注点
- 应核对椭圆对称性与仿射等变性的作用、渐近等价的准确表述，以及置换检验条件和与 Wilks 检验的比较设置。
- 阅读边界
- 索引摘要未注明稿件版本，未给出完整条件或模拟数值；计算便利与稳健性陈述不足以证明在任意分布下表现更好。
原文摘要
New test statistics are proposed for testing whether two random vectors are independent. Gieser and Randles, as well as Taskinen, Kankainen, and Oja have introduced and discussed multivariate extensions of the quadrant test of Blomqvist. This article serves as a sequel to this work and presents new multivariate extensions of Kendall's tau and Spearman's rho statistics. Two different approaches are discussed. First, interdirection proportions are used to estimate the cosines of angles between centered observation vectors and between differences of observation vectors. Second, covariances between affine-equivariant multivariate signs and ranks are used. The test statistics arising from these two approaches appear to be asymptotically equivalent if each vector is elliptically symmetric. The spatial sign versions are easy to compute for data in common dimensions, and they provide practical, robust alternatives to normal-theory methods. Asymptotic theory is developed to approximate the finite-sample null distributions as well, as to calculate limiting Pitman efficiencies. Small-sample null permutation distributions are also described. A simple simulation study is used to compare the proposed tests with the classical Wilks test. Finally, the theory is illustrated by an example.