The Likelihood Ratio Test Of High-dimensional Block Circular Symmetric Covariance Structure

With the rapid development of science and technology,we will inevitably encounter var-ious high-dimensional data when dealing with modern social production problems.For these problems,it is often called "large p,large N" problem because of the large data dimension p and sample size N.In the traditional multivariate statistical hypothesis test,the method of classical chi-square approximation and the LR test can well solve the case where the data dimension is fixed or small.However,the effect of those methods is very poor or even invalid on dealing with high-dimensional data.For this reason,it is a meaningful work to closely contact the production practice and seek new method to resolve the high-dimensional data with "large p,large N"The paper mainly considers the hypothesis test problem with high-dimensional block cir-cular symmetric covariance(BCSC)structure.First of all,for the BCSC structure on the Gaus-sian population,we get the specific expression of moments under the null hypothesis.And then,we consider the approximate distribution of the LR statistics which is more concerned in hypothesis test problems by using the continuity theorem of moment generating function.Here,we adopt two different methods to solve this problem.One is to use the high-order Edgeworth asymptotic expansion(HEE)method to solve the approximate distribution of the LR statistics and conduct error analysis,and the other is to use the high-order Gamma function asymptotic expansion(HGA)method to obtain the asymptotic normality of the LR statistics and give the moderate deviation principle(MDP).At last,the paper gives the comparison of our HGA and HEE methods and the TCA method by some numerical simulation.Under the same parameters,we draw frequency distribution histogram of three methods and list the size values for the same significance level respectively.The chart data show that the TCA method is no longer valid in high-dimensional case,but the proposed HGA and HEE methods perform well.At the same time,the proposed HEE method can control the precision of the approximate distribution by controlling the parameters s;the HGA method has a concise expression and we can observe the convergence rate through MDP.