Moderate Deviation Principle Of Likelihood Ratio Test Statistic For High-dimensional Matrices

In the past decades,most hypothesis testing problems were studied with a fixed sample dimension or a small ratio of sample dimension to sample size 9)due to technological immaturity.However,with the development and progress of productivity,more and more high-dimensional data are being generated,such as genetic data,financial data,medical data,multimedia data,etc.These data have the property that the sample dimension increases with the sample size 9),and the limit of /9)does not tend to 0.Nowadays,high-dimensional data has become a popular topic in various fields.When in high-dimensional cases,more and more hypothesis testing problems have been receiving attention.For instance,hypothesis testing problems for mean vectors and covariance matrices in high-dimensional normal population.When the sample dimension increases with the sample size 9),the limiting distribution of the likelihood ratio test statistic for the above hypothesis testing problem is fully investigated.By comparing with the classical result(when the sample dimension is fixed),it is found that when both the sample size 9)and the sample dimension are large,the traditional research methods are no longer applicable to high-dimensional data,and therefore more and more scholars have started to study high-dimensional data extensively and deeply.Based on the above discussion,this thesis presents the theoretical results related to the likelihood ratio test statistic in the high-dimensional case based on the central limit theorem of the likelihood ratio test statistic,the higher-order Gamma function expansion and the moment expression of the likelihood ratio test statistic,as follows:Chapter 2 investigates the moderate deviation principle of the likelihood ratio test statistic for high-dimensional regression parameter matrices.In this chapter,we consider the hypothesis testing problem on the high-dimensional regression parameter matrix.Based on the central limit theorem of the likelihood ratio test statistic,the moderate deviation principle of the likelihood ratio test statistic is proved under the null hypothesis of the hypothesis testing problem.The rejection region is constructed using the likelihood ratio test statistic,and the Type I error tends to 0 at an exponential rate by using its moderate deviation principle.Meanwhile,numerical simulations verify the accuracy of the moderate deviation principle results.Chapter 3 presents the moderate deviation principle for the likelihood ratio test statistic of high-dimensional mean vectors and covariance matrices.In this chapter,two hypothesis testing problems are investigated.One is to test whether the high-dimensional normal population covariance matrix is arbitrary,and the other is to test whether the mean vector and covariance matrix are arbitrary.For the above two hypothesis testing problems,the moderate deviation principles of the likelihood ratio test statistic are given in this chapter,which holds under both the null and alternative hypotheses of the hypothesis testing problems and is a generalization of the existing results.In addition,the convergence speed of the statistic can be observed by applying the moderate deviation principle.The accuracy of the moderate deviation principle is verified by numerical simulations.Chapter 4 studies the moderate deviation principle of the likelihood ratio test statistic for high-dimensional block compound symmetric matrices.In this chapter,we consider the hypothesis testing problem of a high-dimensional normal population covariance matrix with a block compound symmetric structure.The moderate deviation principle of the likelihood ratio test statistic is obtained under the null hypothesis in the test problem,based on the asymptotic normality of the likelihood ratio test statistic.The results in this chapter are more general than the existing conclusions,and the rate of convergence of the Type I error can be obtained by constructing the rejection region with the likelihood ratio test statistic.In addition,numerical simulations verify the validity of the moderate deviation principle.Chapter 5 analyzes the moderate deviation principle of the likelihood ratio test statistic for high-dimensional correlation coefficient matrices.In this chapter,we test whether the high-dimensional normal population correlation coefficient matrix is diagonal,and proves the moderate deviation principle of the determinant values of the sample correlation coefficient matrix under some conditions.The results hold under both the null and alternative hypotheses in the hypothesis testing problem and is a generalization of existing findings,which greatly enriches the results in this area.