Test On Linear Combinations Of Mean Vector And Covariance Matrix In High Dimensional Data

In the traditional multivariate statistical analysis,it is generally assumed that the sample size n and the dimension p are both finite.However,we often encounter high-dimensional data in which the dimension p and the sample size n are very large,the phenomenon is called "large p,large n" in statistical literature.The classical multivariate analysis method is very poorly or even ineffective in dealing with high-dimensional data.So it will be of great significance to find an effective way to deal with these high-dimensional data.The paper studies the hypothesis test of high-dimensional data when the dimension p and the sample size n tend to be infinity.The first question considers the hypothesis test on whether the mean vectors of k high-dimensional population satisfy the linear combination relation.In the results of Li et al.[21].besides some basic assumptions,the condition of eighth order moment independence for each component Zij of the sample should be satisfied.The results excludes some commonly used distributions,for example,multivariate t distribution,multivariate elliptic distribution and so on.In the paper,we will consider removing the condition,and prove the same result as Li et al.[21]is obtained under weaker conditions,this is to say,the test statistics obeys normal distribution asymptotically under the high-dimensional hypothesis,and the numerical simulations show that-the improved method in this paper is suitable for the normal distribution,multivariate t distribution and so on.Therefore,our results can deal with the more general problem of testing linear combinations of high-dimensional population mean vectors.The other question considers the hypothesis test for the linear relation of the high-dimensional covariance matrix.The paper studies the problem of testing on whether the covariance matrices of k high-dimensional population satisfy the linear combination relation,and we build test statistics.By martingale difference center limit theorem,we also prove that the test statistics obeys normal distribution asymptotically under high-dimension assumption.Finally,some numerical simulations are carried out to verify the efficiency of our test method.