| With the rapid development of computer science and technology,a large amount of high-dimensional data has emerged in modern scientific fields,such as medicine,biology,psychology and finance etc.Classical statistical methods are often derived on the basis that the data dimension p is fixed while the sample size n tends to infinity,or the data dimension is much smaller relative to the sample size.For high-dimensional data,some traditional statistical methods may become inefficient or even ineffective.Therefore,there is a need to find new statistical methods suitable for high-dimensional data.Statistical inference based on covariance matrix plays an important role in mul-tivariate statistical analysis.Based on this,this paper will study the high-dimensional hypothesis testing problems related to population covariance matrix.First of all,this paper studies the hypothesis testing problem of high-dimensional compound symmetric covariance matrix under the assumption of independent compo-nent structure.Based on the Frobenius norm of the difference between two estimates of the population covariance matrix,this paper puts forward four testing methods suit-able for dense and sparse alternative hypotheses.Under the conditions where the data dimension p and sample size n tend to infinity proportionally,i.e.,y_n=p/n→y∈(0,∞),and under the assumption of independent component structure,this paper de-rives the asymptotic distributions of the new test statistics using large-dimensional random matrix theory.Extensive numerical simulations and real data analysis show that,compared to the existing testing methods,the testing methods proposed in this paper have higher powers under different levels of sparsity in alternative hypotheses.Secondly,this paper studies the hypothesis testing problem of high-dimensional compound symmetric covariance matrix under the assumption of elliptical structure.Based on the Frobenius norms of the difference and ratio between two estimates of the population covariance matrix,this paper puts forward five new testing methods.Under the conditions where the data dimension p and sample size n tend to infinity propor-tionally,i.e.,y_n=p/n→y∈(0,∞),and under the assumption of elliptical structure,this paper derives the asymptotic distributions of the new test statistics.These testing methods are not only applicable to high-dimensional data,but also suitable for non-normal populations,demonstrating their universality.Extensive numerical simulations indicate that the testing methods proposed in this paper not only effectively control the probability of Type I errors but also exhibit higher powers under different alternative hypotheses.Finally,this paper addresses the problem of hypothesis testing for high-dimensional compound symmetric covariance matrix with online data.For the testing method of the Frobenius norm of the difference between the two estimates of the population co-variance matrix under the independent component structure,this paper proposes an online data algorithm to enhance the computational speed of online data.Extensive simulation studies reveal that,with the increasing of the data dimension p and sample size n,the proposed online data algorithm proposed in this paper reduces computation time and improves computational speed. |
