Covariance matrix testing for high dimensional data is a fundamental problem. A large class of covariance test statistics based on certain averaged spectral statistics of the sample covariance matrix are known to obey central limit theorems under the null. However, precise understanding for the power behavior of the corresponding tests under general alternatives remains largely unknown. This paper develops a general method for analyzing the power behavior of covariance test statistics via accurate non-asymptotic power expansions. We specialize our general method to two prototypical settings of testing identity and sphericity, and derive sharp power expansion for a number of widely used tests, including the likelihood ratio tests, Ledoit-Nagao-Wolf's test, Cai-Ma's test and John's test. The power expansion for each of those tests holds uniformly over all possible alternatives under mild growth conditions on the dimension-to-sample ratio. Interestingly, although some of those tests are previously known to share the same limiting power behavior under spiked covariance alternatives with a fixed number of spikes, our new power characterizations indicate that such equivalence fails when many spikes exist. The proofs of our results combine techniques from Poincar\'e-type inequalities, random matrices and zonal polynomials.
翻译:对高维数据的共变矩阵测试是一个根本性问题。 根据样本共变矩阵的某些平均光谱统计,已知有大量基于样本共变矩阵中某些平均光谱统计的共变测试统计数据,可以遵守空下中央限制的理论。 但是,对普通替代品下相应测试的实力行为的精确理解仍然基本上未知。本文开发了一种分析共变测试统计数据通过精确的非无损功率扩展进行的力量行为的一般方法。我们把一般方法专门用于两种测试身份和外观的原变异设置,并获得一系列广泛使用的测试的急剧功率扩张,包括概率比测试、莱多-纳高-沃尔夫测试、凯-马测试和约翰测试。这些测试的功率扩张在所有可能的替代物的温度增长条件下,在维度至模比的微增速比率下,对所有可能的共变异性统计方法都保持统一。有趣的是,虽然这些测试中有些以前知道在加压的共变异性替代品下,有固定数量的峰值,但我们的新功率特征描述显示,当存在许多峰值时,这种等同性技术会综合了我们多级的多元性。