Homogeneity Test Of K Covariance Matrices For Large-dimensional Data

In recent decades,statistical analysis of large-dimensional data has been widely incorporated in various fields.The study of large-dimensional covariance matrices has been attracting more and more attention recently.However,the sample covariance matrix will degenerate if the dimension of the sample data is larger than the sample size.In this situation,traditional test tools are no longer valid;new approaches are needed to solve these problems.This paper focuses on three testing problems: the covariance matrix is equal to a given matrix,the covariance matrix is proportional to a given matrix,and the equality of K large-dimensional covariance matrices,for K>2.Firstly,we propose the correct likelihood ratio test and the locally most powerful invariant test for the large-dimensional covariance matrix.Then we compare the size and power of these two methods.Secondly,we apply them to test the covariance matrix of the colon data.In this paper,we study the equality of K(K >2)large-dimensional covariance matrices and propose a method of multiple comparisons without the normality assumption.In this method,we convert the original hypothesis testing of K covariance matrices into hypothesis testing of mutual equality of K-1 covariance matrices.We then test this problem by applying the test statistics proposed by Bai et.al(2009),Li and Chen(2010)and Ahmad(2017).The method of multiple comparisons can be further applied to hypothesis testing of multi-linear combinations of covariance matrices,in special cases.We propose a new test statistic and derive its asymptotic distribution.One can also use the principal component analysis when a few of the eigenvalues of the covariance matrices are far larger than the other eigenvalues.The simulation results demonstrate that the proposed tests are valid and significantly better than traditional tests.Finally,we apply the newly proposed methods to test whether the covariance matrices of four isoforms of gene expression in small round blue cell tumors(SRBCT)data set are equal.The p-value of these tests are all zero and this implies that any assumption of equality of these four tumor isoforms’ covariance matrices is false.