Likelihood Ratio Tests For High-dimensional Covariance Matrices And Correlation Matrices

With the development of science and technology,the storage capacity of computers has gradually enhanced,and the high-dimensional data from all walks of life can be saved.For high-dimensional data with a large dimension relative to the sample size,many classical statistical methods may perform poorly or even fail,which brings great challenges to statisticians.To effectively process these high-dimensional data,it is necessary to generate applicable statistical methods.The covariance matrix and correlation matrix play an important role in multivariate statistical analysis and are common matrices in principal component analysis,discriminant analysis,and factor analysis.Statistical inference of covariance matrix and correlation matrix is a problem to be solved in multivariate statistical analysis.Based on this,this paper will study the likelihood ratio test of high-dimensional covariance matrix and correlation matrix.First of all,this paper tests whether a population covariance matrix is equal to the identity matrix and whether two population covariance matrices are equal.Combining likelihood ratio statistics and the statistics based on the Frobenius norm,the new statistics are proposed.The new test method is suitable not only for low-dimensional data,but also for high-dimensional data.Numerical simulation shows that the new test method can well control the type I error rate for both Gaussian and non-Gaussian distributions,and has high empirical power for sparse and dense alternative hypotheses.Secondly,this paper tests whether a population correlation matrix is equal to a given matrix.Similarly,Combining likelihood ratio statistics and the statistics based on the Frobenius norm,the new statistics is proposed.In this paper,the likelihood ratio statistic is extended to the case that the dimension p can be greater than or equal to the sample size n.Under the null hypothesis,the asymptotic properties of the new test method are obtained by using large dimensional random matrix theory when the data dimension and sample size tend to infinity proportionally.Simulation results show that the proposed test method has good performance for sparse and dense alternative hypotheses.Finally,this paper studies the likelihood ratio test of the high-dimensional covariance matrix.This paper describes the phenomenon that the empirical power increases first,then decreases,and then increases when the likelihood ratio statistics are used to test the problem of high-dimensional population covariance matrix equal to the identity matrix,and analyzes the reasons for this phenomenon.The improved likelihood ratio statistic is proposed for addressing this phenomenon,and its asymptotic distribution is given under the null hypothesis with its empirical power analyzed.Numerical simulation shows that when the dimension p is close to the sample size n,the power of the modified likelihood ratio statistic does not decline.