Testing The Structures Of High-dimensional Covariance Matrices

With the rapid development and wide applications of computer techniques,high-dimensional data can be collected and stored. This is called as high-dimensionaldata or large-dimensional data(see [7]). Many traditional estimation and test toolsare no more valid or perform badly for such high-dimensional data, since thesetraditional methods are often based on the classical central asymptotic theoremswhich assuming a large sample size and fixed dimension. Therefore, some newstatistical methods about high-dimensional data analysis have been studied in thelast score years,(see [3],[6],[5],[12], etc.). Especially, the random matrix theory(RMT)[7] and the central limit theorems (CLT,[60]) for linear spectral statisticsplay important roles in the statistical inference.In the second chapter of this article, we consider testing proportionality ofhigh-dimensional covariance matrices from two diferent populations. The propor-tionality of covariance matrices is the simplest form of heteroscedasticity betweenpopulations, which has extensive applications in economics, discriminations, etc.This paper generalizes the work of [5] and concerns the test of proportionality oftwo high-dimensional covariance matrices Σ1and Σ2which allows any positiveconstant. Based on the modern random matrix theory, this paper proposes apseudo-likelihood ratio test (PLRT) and proves asymptotic normality property asthe dimension and sample sizes (1,2) tend to infinity proportionally. Sim-ulation studies show that the pseudo-likelihood ratio test behaves well for bothhigh-dimensional Gaussian and non-Gaussian distributions.The independence test for two multivariate variables is a classical testing prob-lem in multivariate statistical analysis and also widely used in real life. For instance,in canonical correlation analysis, it is important to know whether two sets of vari-ates are independent. Besides, in micro-array data analysis on genes and DNA test,it is meaningful to check whether there is correlation among pieces of genes. The fourth chapter of this paper proposes a new test procedure by trace criterion fortesting the independence of sets of two large-dimensional multivariate variables. Anew test statistic is developed based on the modern random matrix theory whichbehaves well no matter the dimension is either small or large relative to the samplesize. Under some regularity conditions, the asymptotic normality property of thenew test statistic as the dimensions and the sample size tend to infinity simultane-ously and proportionally is established. Numerical simulations are carried out toillustrate that the new test statistic is more efective then existent ones.