One-sample And Two-sample Testing Methods For High-Dimensional Covariance Matrices

With the arrival of the era of big data,the dimension of many data is not only large relative to the sample size,but the structure of the data also becomes complicated.For such high-dimensional data,many classical statistical methods may perform poorly or fail.Therefore,it is of great significance to find effective methods to analyze these high-dimensional data.Covariance matrix is one of the important tools for analyzing high-dimensional data.In the fields of economics,biology and etc.,many methods of processing data require to make statistical inference on covariance matrix.For this reason,this paper will study the high-dimensional hypothesis testing problems related to population covariance matrix.First of all,this paper studies the hypothesis testing problem of the equality of two high-dimensional population covariance matrices.Based on the difference and ratio of two sample covariance matrices,this paper proposes three testing methods for dense alternative hypothesis and sparse alternative hypothesis: one testing method is shown to be powerful against dense alternative hypothesis,and the other two testing methods are suitable for general cases,including dense alternative hypothesis,sparse alternative hypothesis,or a mixture of the two.Based on large-dimensional random matrix theory,the asymptotic properties of these three testing methods are derived under the condition that the data dimension and sample sizes tend to infinity proportionally and the null hypothesis.In addition,the asymptotic power function of these three testing methods are also studied under the representative alternative hypotheses.Extensive simulation studies demonstrate that,compared with the existing testing methods,the proposed testing methods in this paper have good performance in terms of the type I error rate and the empirical power.Secondly,this paper proposes a testing method to test whether high-dimensional data comes from the stationary vector autoregressive process,and this testing method also strictly relies on the sample covariance matrix.The vector autoregressive process considered in this paper is a special kind of vector autoregressive process,in which thecoefficient matrices are diagonal with the same diagonal elements.Therefore,under the null hypothesis,the data can be regarded as an independent and identically distributed sample from a stationary autoregressive process.In this paper,the least square estimation method of unknown coefficients in the autoregressive process is given,and the asymptotic normality of the estimation is derived.In addition,based on the covariance matrix of the autoregressive process and the estimation of unknown coefficients,a new testing method is proposed to test whether a set of data comes from the abovementioned stationary vector autoregressive process.This testing method is not only suitable for high-dimensional data but also for low-dimensional data.In particular,the testing method is universal in the sense that it makes no assumption on distribution.Simulation studies show that the testing method performs well under different models and different parameter settings.