Adaptive Tests And Change-Point Detection Of High-Dimensional Covariance Matrices

Classical statistical methods were developed under the assumptions that the data dimension p remains fixed and the sample size n tends to infinity,or that p is relatively small compared to n.But high-dimensional data often involve a large number of feature variables,where p is large or even larger than n.In such cases,some classical statistical methods may no longer be suitable for high-dimensional data.Therefore,it is necessary to propose appropriate statistical methods for the analysis of high-dimensional data.As known to all,covariance matrix is one of the key characteristics of the popula-tion distribution,and hypothesis testing problems related to it have received widespread attention in the field of multivariate statistical analysis.This paper delves into three s-tudies regarding high-dimensional covariance matrices:first,the hypothesis test on test-ing that a high-dimensional covariance matrix has a block-diagonal structure;second,the hypothesis test on the homogeneity of several high-dimensional covariance matri-ces;third,the change-point detection of several high-dimensional covariance matrices.Concerning the above three studies,this paper utilizes the construction of U-statistic to develop some statistics based on a family of l_a-norms,and proposes corresponding adaptive tests and estimation methods.The proposed approaches are applicable not only to high-dimensional populations but also to non-Gaussian populations.New tests in this paper are widely adapted to the alternatives with different sparsity levels(e.g.,extreme-ly sparse,relatively sparse,relatively dense,dense,etc.)and the proposed estimators perform well in the detection of single and multiple change-points of high-dimensional covariance matrices.Firstly,for testing that a high-dimensional covariance matrix has a block-diagonal structure,this paper constructs a family of U-statistics based onl_a-norms of interest-ed parameter vector and establishes the joint asymptotic normality and asymptotic in-dependence of these U-statistics under the null and alternative hypotheses,with some regular conditions satisfied.To leverage the advantages of the tests based on different U-statistics,an adaptive test statistic is developed by using the minimum p-value criterion.Furthermore,under general alternative hypothesis,this paper investigates the asymptot-ic power functions of the tests based on the constructed U-statistics and the proposed adaptive test statistic.Extensive simulation studies demonstrate that the proposed tests have good performances in controlling the type I error rate,and that the adaptive test could maintain the high powers against the alternatives with a wide range of sparsity levels.Secondly,for testing the homogeneity of several high-dimensional covariance ma-trices,this paper constructs a series of U-statistics based onl_a-norms of interested pa-rameter vector.Furthermore,under the null and local alternative hypotheses,the joint asymptotic normality and asymptotic independence of these U-statistics are derived un-der certain regular conditions.Building upon above asymptotic results,a family of max-l_a-type statistics are put forward by using the maximization idea.To combine the advantages of the tests based on different max-l_a-type statistics,two adaptive test statistics are proposed by using the minimum p-value criterion and Fisher’s method,re-spectively.Numerical simulations show that the proposed adaptive tests could maintain the high powers against the alternatives with various sparsity levels.Finally,for detecting the change-points of several high-dimensional covariance matrices,this paper proposes some estimators of single change-point based on the theo-retical results established for the homogeneity test of high-dimensional covariance ma-trices,and derives the convergence rates of these estimators.Moreover,three adaptive estimators are developed.To detect multiple change-points of high-dimensional covari-ance matrices,this paper builds the corresponding adaptive estimators of the number and locations of change-points by introducing the binary segmentation algorithm,and proves the consistency of these estimators.Numerical simulations show that the pro-posed adaptive estimators perform well in the detections of single and multiple change-points of high-dimensional covariance matrices.