Testing Two Types Of High-dimensional Matrices

In the era of big data,more and more attention has been paid to the statistical inference of high-dimensional data.While big data has been increasingly applied in finance,medical treatment,biology and other fields,processing such high-dimensional data also brings us huge challenges.Many traditional statistical inference methods assume that dimension of the data p is fixed and the sample size n goes to infinity.These methods are even invalid facing high-dimensional data(the case that p is larger compared with n),so we need to propose new methods to study such ultrahigh-dimensional data.In this thesis,hypothesis testing procedures are proposed for two different types of high-dimensional matrices.First,we propose two new statistics to test the identity of high-dimensional covariance matrix.Applying the large dimensional random matrix theory,we study the asymptotic distributions of our proposed statistics under some mild conditions.The proposed tests can accommodate the situation that the data dimension is much larger than the sample size,with no underlying distributional assumptions.Numerical studies demonstrate that the proposed tests have good performance on the empirical powers when the dimension p and sample size n tend to infinity proportionally.Our second testing procedure is on the coefficient matrix in a high-dimensional matrix regression model.In the high dimensional matrix linear regression model,we study the relationship between the high dimensional response matrix and the scalar covariates,where the dimension of the coefficient matrix is p × q.To determine whether the covariates have effects on the matrix response,we want to test whether the coefficient matrices are equal to the zero matrix.For this reason,we propose a test statistic and study its asymptotic distribution.Numerical simulation shows our results are pretty good.Finally,we use the multiple testing method to screen the covariates in the matrix regression model.The results show that our method can quickly reduce the dimension without losing the important variable information.In the two test questions mentioned above,the dimension p,q and the sample size n tend to infinity proportionally,and our test method does not depend on the distribution of samples.