Theoretical Research And Application Of Spatial Signed Covariance Matrix And Spatial Median Under High-dimensional Ellipsoid Mode

This paper considers the asymptotic properties of the eigenvectors of the spatial sign covariance matrix(SSCM),the eigenvalues of the spatial sign Fisher matrix(SSFM),and the spatial median under elliptical distributions in a highdimensional framework.The high-dimensional framework considered here is that the dimension of observations and the sample size 9)both diverge to infinity with their ratio tending to a constant (8.In Chapter 2,we study the theoretical properties of the eigenvector empirical spectral distribution(VESD)of the sample SSCM in a high-dimensional framework.Under some conditions,we first prove that the limiting spectral distribution of the VESD is a generalized Marˇcenko-Pastur distribution.Secondly,a central limit theorem of the linear spectral statistics defined by the VESD function is established.And then conclude that the eigenvector matrix of the sample SSCM is asymptotically Haar distribution when the shape matrix is an identity matrix under elliptical distributions.Finally,some simulations are conducted to illustrate the theoretical results.In Chapter 3,we define the sample SSFM and then study the asymptotic properties of its eigenvalues,including:(1)the limiting spectral distribution of the empirical spectral distribution(ESD)of the sample SSFM;(2)the central limit theorem of the linear spectral statistics defined by these eigenvalues.According to the above results,we construct a new statistic to test whether the two population covariance matrices are proportional under elliptical distributions.Finally,simulations show that the performance of the proposed test statistic is superior to the compared methods under some settings.In Chapter 4,we discuss the asymptotic behaviors of the spatial median under elliptical distributions in a high-dimensional framework.Firstly,we give a new asymptotic expression of the sample spatial median,and then study the first and second-order asymptotic limits of the Euclidean distance between the sample spatial median and its population counterpart.Based on these findings,new one-sample and two-sample test procedures for high-dimensional mean vectors are developed.In the end,the test procedures are compared with other methods through some simulations.