Asymptotic normality and moderate deviation principle for high-dimensional likelihood ratio statistic on block compound symmetry covariance structure

The paper considers a high-dimensional likelihood ratio (LR) test on the block compound symmetric (BCS) covariance structure of a multivariate Gaussian population. When the dimension of each block p, the number of blocks u and the sample size n satisfy that and pu<n-1 as , the asymptotic normality and the moderate deviation principle of the logarithmic LR statistic are obtained. Some numerical simulations demonstrate that the proposed method in high-dimensional BCS test outperforms the traditional Chi-square approximation method, and it is as efficient as the Edgeworth expansion method by Mitsui et al. [Likelihood ratio test statistic for block compound symmetry covariance structure and its asymptoic expansion. Technical Report No.15-03, Statistical Research Group, Hiroshima University, Japan; 2015]. In addition, the proposed method is more applicable because the asymptotic distribution of the test statistic is more concise and the restriction on parameters p, u is milder.