Limiting Properties Of Outlier Eigenvalues Of Sample Covariance Matrix Under Large-dimensional Mean Mixture Mode

Spiked population model is an important theoretical model in high-dimensional statistical analysis.Its specific performance is that under certain conditions,if there are some spiked eigenvalues in the population covariance matrix,the corresponding sam-ple covariance matrix will also carry spiked eigenvalues.Different from the traditional Spiked population model in which the sample spiked eigenvalues come from the pertur-bations of the population covariance matrix,sample covariance matrices from a finite mean mixture model naturally carry certain spiked eigenvalues,which are generated by the differences among the mean vectors.However,their asymptotic behaviors remain largely unknown when the population dimension p grows proportionally to the sample size n.This paper mainly studies the asymptotic properties of the spiked eigenvalues of the sample covariance matrix from mean mixture model under the high-dimensional framework.This includes the central limit theorem for these spiked eigenvalues,under both Gaussian and non-Gaussian assumptions,and the asymptotic independence of the sample spiked eigenvalues and the linear spectral statistics.And the research on the spiked eigenvalues of the mean mixture model is extended to the joint distribution re-search of the linear spectral statistics and the random sesquilinear forms under the mean mixture model.The first chapter is an introduction,which mainly introduces the background sig-nificance,difficulties and innovations of our research.And some necessary preliminar-ies of large-dimensional random matrix theory and Spiked population model are given.In the second chapter,under the Gaussian assumption,we give the asymptotic properties of the sample spiked eigenvalues under the Gaussian mean mixture model.Here we establish a new central limit theorem different from the traditional Spiked popu-lation model for the sample spiked eigenvalues.The results show that the convergence rate of these spiked eigenvalues is O(1/(?))and their fluctuations can be character-ized by the mixing proportions,the eigenvalues of the common population covariance matrix,and the inner products between the mean vectors and the eigenvectors of the population covariance matrix.In the third chapter,under the general distribution,we give the convergence of the sample spiked eigenvalues under the mean mixture model.Here we study the joint distribution of three kinds of random sesquilinear forms composed of the sample mean,the populations mean and the sample covariance matrix under the mean mixtue model.These random sesquilinear forms constitute many famous statistics in multivariate sta-tistical analysis,such as T~2statistics,Regularized T~2statistics,multivariate analysis of variance statistics.And the sample spiked eigenvalues of the mean mixture model we studied are also determined by these random sesquilinear forms.Here we establish the joint central limit theorem for these random sesquilinear forms and as an application of this theoretical result,we generalize the results of Chapter 2 to develop the central limit theorem for the sample spiked eigenvalues of mean mixture model under the general dis-tribution.This result indicates that the second-order convergence of the sample spiked eigenvalues is only related to the fourth-order moment of the population distribution.In chapter four,under the general distribution,we give the asymptotic indepen-dence of sample spiked eigenvalues and linear spectral statistics under mean mixture model.Here we study the central limit theorem of linear spectral statistics under mean mixture model,and find that they are asymptotically independent of the three kinds of random sesquilinear forms we studied in chapter three.As an application of this theory,we show that under the mean mixture model,the(normalized)sample spiked eigenvalues are asymptotically independent of the(normalized)linear spectral statis-tics.At the same time,since the test statistics of the covariance matrices are usually composed of linear spectral statistics,and the test statistics of the mean vectors are usu-ally composed of random sesquilinear forms,using the asymptotic independence of the two,we propose new test statistics for the simultaneous test of covariance matrices and mean vectors under multiple populations.Simulation results show that the finite sample performance of the proposed test statistics is quite good.The fifth chapter is the summary of the full paper,and gives several extension directions of this research.