A high-dimensional likelihood ratio test for circular symmetric covariance structure

The paper considers a high-dimensional hypothesis test on circular symmetric covariance structure. When both the dimension p and the sample size N tend to infinity with pN→y∈(0,1]$\frac{p}{N}\rightarrow y\in (0,1]$, it proves that under the assumption of Gaussian, the logarithmic likelihood ratio statistic converges in distribution to a Gaussian random variable, and the specific expressions of the mean and the variance are also obtained. The simulations indicate that our high-dimensional likelihood ratio method outperform those of traditional chi-square approximation method and high-dimensional edgeworth expansion method, and it is as effective as the more accurate high-dimensional edgeworth expansion method on analyzing the circular symmetric covariance structure of high-dimensional data.