High-dimensional Edgeworth expansion of LR statistic for testing block circular symmetry covariance structure and its errors

The paper considers the asymptotical expansion on the likelihood ratio (LR) statistic for testing the block circular symmetric (BCS) covariance structure of a multivariate Gaussian population. When the number of blocks u and the dimension of each block p satisfy p - p(n) ->infinity and pu/(n -1) -> c is an element of (0,1) as the sample size n -> infinity, the Edgeworth expansion of the null distribution of the LR test statistic and its uniform Berry-Esseen type bound are established. Some numerical simulations indicate that the proposed approximation is more accurate than the traditional Chi-square approximate method on dealing with the high-dimensional test.