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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  satisfyandas the sample sizethe 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.

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