Structural Test Of Covariance Matrix In Large Dimensional Framewor

With the advent of the era of big data and the development of computer technology,more and more high-dimensional data are encountered in various fields.According to this phenomenon,some of the multivariate procedures which are established under the classical framework(assuming the dimension p is fixed,sample size n→∞)may not necessarily work.Thus,new statistical methods ought to be proposed so as to handle the high-dimensional situation.This paper focuses on some tests for high-dimensional covariance matrices under the large dimensional framework(assuming that n,p→∞together with p/n→ c∈(0.∞)).The covariance matrix ∑ is a most important random matrix in multivariate statistical inference.It is fundamental in hypothesis testing,principal component analysis,factor analysis,and discrimination analysis.Many test statistics are defined by its eigenvalues.In this paper,we consider testing the structure(identity/sphericity/diagonal)of high-dimensional matrices.The practical needs for testing the above hypotheses come from several areas of statistical applications.In specific,it is common to assume the gene-wise independence in the expression levels.As a consequence,taking testing procedures to ascertain such assumptions can make subsequent studies more reliable.This paper focuses on topics about testing whether the covariance matrix ∑ equals to the identity matrix or owns spherical/diagonal structure.Previous test statistics can be grouped into three categories.Likelihood ratio statistics,sum-type statistics(e.g.based on the Frobenius norm of the differences between ∑ and its corresponding form under the null hypothesis),and max-type statistics(e.g.based on the maximal element of the differences between ∑ and its corresponding form under the null hypothesis).As we know,the likelihood ratio statistics are well defined only when p<n,and thus are not suitable for p≥ n situations.In particular,the sum-type tests are powerful against dense alternatives,and will lose their superiority and even become inconsistent against sparse alternatives.However,the max-type tests are in favor of sparse alternatives.In conclusion,for the above hypotheses,we pay attention to establishing new tests which can accommodate both dense alternatives and sparse ones.It is worth mentioning that,our tests are based under general independent components models,which are more general than Gaussian model.The first chapter is the introduction,which mainly expounds the influence of highdimensional data on conventional statistical theory,and gives the preliminary knowledge,model settings and assumptions required for the content of the subsequent chapter.The second chapter is about a test for the identity of a high-dimensional covariance matrix,that is,testing that whether ∑=I,where I represents the identity matrix.Based on (?) and (?),we propose a new test statistic Tid.Under the independent component model with finite fourth-order moments,we further establish the central limit theorem of this statistic when the null hypothesis holds.By assuming that the matrix under the alternative hypothesis satisfies a certain sparse structure,the asymptotic distribution of the statistic is also proved.The simulation results show that the test statistic Tid we constructed is indeed compatible with sparse and dense preparations,and outperforms other existing test statistics for many cases.The third chapter considers a test for the sphericity of a high-dimensional covariance matrix,that is,testing that whether ∑=σ2I,where σ2 is an unknown but finite positive constant.Similar to the identity matrix test,using tr(Sn)/p as to estimate the parameter σ2,we construct a new test statistic based on (?) and(?).Under the independent component model with finite fourth-order moments,we further establish the central limit theorem of this statistic when the null hypothesis holds.By assuming that the matrix under the alternative hypothesis satisfies a certain sparse structure,the asymptotic distribution of the statistic is also proved.The simulation results show that the test statistic Tsp we constructed is compatible with sparse and dense preparations,and outperforms other existing test statistics in intermediate states under many cases.The fourth chapter concerns a test for the covariance matrix to be diagonal,that is,for any diagonal matrix Λ,testing that whether ∑=Λ.Based on the summation of the quartic of the upper half-diagonal elements from the sample covariance matrix,i.e.,(?),we construct a new test statistic Tdiag.Under the assumption of an independent component model with finite fourth-order moments,we further provide the central limit theorem of the statistic when the null hypothesis holds.By assuming that the matrix under the alternative hypothesis satisfies a certain sparse structure,the asymptotic distribution of the statistic is also proved.The simulation results show that the diagonal test statistic we constructed can indeed accommodate both sparse and dense alternatives,and outperforms other existing test statistics in intermediate states that are neither sparse nor dense.Chapter 5 summarizes the results of the full text,and gives the direction of further development of this research.