Largest entries of sample correlation matrices from equi-correlated normal populations
Largest entries of sample correlation matrices from equi-correlated normal populations
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DOI：
10.1214/19-aop1341
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发表时间：
2019-09
期刊：
影响因子：
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通讯作者：
Jianqing Fan;Tiefeng Jiang
中科院分区：
文献类型：
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作者：
Jianqing Fan;Tiefeng Jiang
研究了具有等相关结构的高维高斯总体样本相关矩阵最大非对角元的极限分布。假设总体分布的项具有共同的相关系数ρ > 0，并且总体维度p和样本量n都趋于无穷大，logp = o（n 3）。当0 <p < 1时，我们证明了样本相关矩阵的最大非对角项收敛于高斯分布，当0 <p < 1/2时，样本协方差矩阵也是如此。这与一个众所周知的独立情形下的结果有很大不同，在独立情形下，ρ = 0，上述极限分布是极值分布。然后，我们研究这两个极限分布之间的相变，并确定过渡发生的ρ制度。如果ρ小于、大于或等于阈值，则相应的极限分布分别是极值分布、高斯分布以及两个分布的卷积。证明依赖于陈斯坦泊松近似方法，条件，耦合创建独立性和样本相关矩阵的特殊属性的微妙使用。最后给出了一个统计检验问题的应用。
The paper studies the limiting distribution of the largest off-diagonal entry of the sample correlation matrices of high-dimensional Gaussian populations with equi-correlation structure. Assume the entries of the population distribution have a common correlation coefficient ρ > 0 and both the population dimension p and the sample size n tend to infinity with logp = o(n 3 ). As 0 < ρ < 1, we prove that the largest off-diagonal entry of the sample correlation matrix converges to a Gaussian distribution, and the same is true for the sample covariance matrix as 0 < ρ < 1/2. This differs substantially from a well-known result for the independent case where ρ = 0, in which the above limiting distribution is an extreme-value distribution. We then study the phase transition between these two limiting distributions and identify the regime of ρ where the transition occurs. If ρ is less than, larger than or is equal to the threshold, the corresponding limiting distribution is the extreme-value distribution, the Gaussian distribution and a convolution of the two distributions, respectively. The proofs rely on a subtle use of the Chen–Stein Poisson approximation method, conditioning, a coupling to create independence and a special property of sample correlation matrices. An application is given for a statistical testing problem.