Testing The Homogeneity Of Two High-dimensional Population Covariance Matrices

Multivariate statistical analysis(MSA)is a method to deal with agronomic,med-ical,ergonomic,meteorologic,geological,psychological problems.It is of great value in practical use.Multivariate statistical analysis is a natural promotion of the unitary statistical analysis.Intuitively,it's easy to understand that observing in several aspect-s of the object is more comprehensive and more accurate than just one-sided aspect.Thus multivariate statistical analysis develop rapidly accompanying to the unitary s-tatistical analysis.Because of the development of the computer technology and the huge size of applied data,especially since the year of 2000,people realize that classi-cal MSA tools can not provide solutions,which are accurate enough within tolerable errors,when dealing with high-dimensional data.Meanwhile the high-dimensional s-tatistical analysis makes a great progress.Homogeneity of the variance is a significant requirement in statistical analysis,especially in linear regression and analysis of vari-ance.The problem turns to be homogeneity of covariance in multivariate statistical analysis.As we talked,it's an significant problem to consider the homogeneity of the covariance in high-dimensional condition,thus we aim to propose a predominant test in this circumstance.In this article,we focus on the problem of the homogeneity of the covariance ma-trices,and make it possible in high-dimensional situation.Chapter 1 is the preliminar-ies,which contains classical method of the testing of the covariance matrices,method in high-dimensional circumstance and how Random Matrix Theory(RMT)deal with covariance testing in high-dimensional situation and basic concepts.In Chapter 2,we propose the Central Limit Theory(CLT)for Linear Spectral Statistics(LSS)to dispose the high-dimensional testing.This is based on modifying the Logarithm Likelihood Ratio Test(Log-LRT)statistic.An advantage of using Log-LRT statistic is the in-variance under linear transformation.Another logarithmic statistic with easier form is proposed and the comparison with Log-LRT statistic,Li and Chen[25]and Cai et al.[14]is also exhibited in Chapter 2.Also,to guarantee our statistics invariable,we give a method to estimate forth moment and prove that the estimator asymptotically unbi-ased.The performance of the estimator is also shown in Chapter 2.In Chapter 3,we modified the Pillai's statistic.Essentially Pillai's statistic is a trace form statistic which leads to computational simplicity.In addition,Pillai's trace statistics can be utilized in the space with p=1 or p/n2=1,due to the linearity of the integrand functions,however,Log-LRT statistic is not well-defined at that situation.The comparison with the four statistics we proposed,Li and Chen's[25]statistic,Cai et al.'s[14]statistic are showed in Chapter 3.In Chapter 4,an application with real data using the proposed statistics is demonstrated.All the proofs are shown in Chapter 5.