High-dimensional consistent independence testing with maxima of rank correlations
High-dimensional consistent independence testing with maxima of rank correlations
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DOI：
10.1214/19-aos1926
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发表时间：
2018-12
期刊：
影响因子：
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通讯作者：
M. Drton;Fang Han;Hongjian Shi
中科院分区：
文献类型：
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作者：
M. Drton;Fang Han;Hongjian Shi
测试高维观测的相互独立性是一个基本的统计挑战。基于线性和简单等级相关性的流行测试已知不能检测非线性、非单调关系，需要能够解释这种依赖性的方法。为了解决这一挑战，我们提出了一个家庭的测试，使用最大值的成对秩相关性，允许一致的成对独立性评估。建立在一个新开发的Cram\'{e} r型中偏差定理退化U统计量，我们的结果涵盖了各种秩相关，包括Hoeffding的$D$，Blum-Kiefer-Rosenblatt的$R$，和Bergsma-Dassios-Yanagimoto的$\tau^*$。所提出的测试是分布免费的类的多元分布连续利润，无需置换的情况下实现，并被证明是速率最优的高斯copula模型下的稀疏替代品。作为研究的副产品，我们揭示了上述三个秩相关统计量之间的同一性，从而向证明Bergsma和Dassios的猜想迈出了一步。
Testing mutual independence for high-dimensional observations is a fundamental statistical challenge. Popular tests based on linear and simple rank correlations are known to be incapable of detecting non-linear, non-monotone relationships, calling for methods that can account for such dependences. To address this challenge, we propose a family of tests that are constructed using maxima of pairwise rank correlations that permit consistent assessment of pairwise independence. Built upon a newly developed Cram\'{e}r-type moderate deviation theorem for degenerate U-statistics, our results cover a variety of rank correlations including Hoeffding's $D$, Blum-Kiefer-Rosenblatt's $R$, and Bergsma-Dassios-Yanagimoto's $\tau^*$. The proposed tests are distribution-free in the class of multivariate distributions with continuous margins, implementable without the need for permutation, and are shown to be rate-optimal against sparse alternatives under the Gaussian copula model. As a by-product of the study, we reveal an identity between the aforementioned three rank correlation statistics, and hence make a step towards proving a conjecture of Bergsma and Dassios.