Abstract With the advent of the era of big data, high dimensional covariance matrices are increasingly encountered and testing covariance structure has become an active area in contemporary statistical inference. Conventional testing methods fail when addressing high dimensional data due to the singularity of the sample covariance matrices. In this paper, we propose a novel test for the prominent identity test and sphericity test based on posterior Bayes factor. For general population model with finite fourth order moment, the limiting null distribution of the test statistic is obtained. Furthermore, we derive the asymptotic power function when the sample size and dimension are proportional against spiked alternatives. When the dimension is much larger than the sample size, under general alternatives, the limiting alternative distribution together with the consistency of the new test is also obtained. Monte Carlo simulation results show that the limiting approximation is quite accurate under the null for finite sample, and the proposed test outperforms some well-known tests in the literature in terms of Type I error rate and the empirical power.

中文翻译：

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通过后验贝叶斯因子测试高维协方差矩阵

摘要 随着大数据时代的到来，越来越多地遇到高维协方差矩阵，检验协方差结构成为当代统计推理的一个活跃领域。由于样本协方差矩阵的奇异性，传统的测试方法在处理高维数据时会失败。在本文中，我们提出了一种基于后验贝叶斯因子的突出身份测试和球形度测试的新测试。对于四阶矩有限的一般总体模型，得到检验统计量的极限零分布。此外，当样本大小和维度与尖峰替代品成比例时，我们推导出渐近幂函数。当维度远大于样本量时，在一般替代方案下，还获得了限制替代分布以及新测试的一致性。蒙特卡罗模拟结果表明，有限样本在零点下的极限近似是相当准确的，并且所提出的测试在I类错误率和经验功效方面优于文献中的一些众所周知的测试。