Hypothesis Testing On High-dimensional Covariance Matrices And Beyond

Random matrices play a key role in multivariate statistical analysis,which is con-cerned with data consist of sets of measurements on a number of objects.Especially,covariance matrices and correlation matrices can be often used in principal compo-nents analysis,factor analysis and discriminant analysis such important statistical area.Therefore,estimation and inference on covariance matrices or correlation matrices ap-pear to be very essential.Anderson(2003)[3]conducted a framework on estimating and testing covariance matrices and correlation matrices,which mostly can be summa-rized as maximal likelihood estimators and likelihood ratio tests,it is well known that those methods have some optimal properties when the data dimension p is fixed.Recently,modern statistical data in scientific fields such as microarray gene ex-pressions in biology are increasingly high dimensional,which refers to the data dimen-sion increases to infinity as the number of observations and even the dimension can be much larger than the sample size.This is not a good phenomenon,as all conventional methods about testing for covariance matrices such as likelihood ratio test may not necessarily work for high-dimensional data in the literature.In this thesis,we propose some new and novel hypotheses testing methods for covariance matrices and correlation matrices,the proposed tests can be both relax to the non-normal populations,as well as can accommodate situations under ultra high dimensionality where the data dimension is much larger than the sample size,namely the "large p,small n" situations.The asymptotic normality is derived under both null hypothesis and alternative hypothesis,empirical studies are demonstrated that the test has good properties for a wide range of dimensions and sample sizes.There are mainly five parts in this thesis.In Chapter 1 we review briefly the background of this thesis and introduce the outline of development of this subject in recent years.Moreover,we introduce the structure of the full thesis and describe the main content of this thesis.and we also review briefly U statistics and also give a discussion on high-dimensional U statistics.In Chapter 2 we propose a new testing statistic on proportionality of two high-dimensional covariance matrices.and derive its asymptotic properties.In Chapter 3 we propose a new testing statistic on banded structure of high-dimensional precision matrices.and derive its asymptotic properties.In Chapter 4 we propose a new testing statistic on equality of two high-dimensional correlation matrices.and derive its asymptotic properties.