The Corrected Rao's Score Test Statistic For Large-dimensional Covariance Structure And Beyond

A significant and constant advancement in the world has been in the rapid development of computer science,which has enabled one to collect and store data sets of very high dimension conveniently.So,high-dimensional data or even large-dimensional(large p small n)data appeared widely in various fields,such as genetic statistics,signal processing,stock market analysis.However,classical statistical meth-ods are under the assumption that the data dimension p is fixed and the data dimen-sion p is more smaller than the sample size n.So classical limiting theorems are not valid for the high-dimensional data.Therefore,in the past decades,many researchers proposed new methods for high-dimensional data,for example,random matrix the-ory.The covariance matrices have a wide application for its simple structure,and the tests of two-sample covariance matrices play significant roles in multivariate analysis Therefore,it is one of the important roles in random matrix theory.This paper pays attention to the tests of covariance matrices,firstly,this paper introduces the corrected Rao's score test statistic for the large-dimensional covariance structure proposed by Jiang(2016),and simulation study shows that this statistic can provide better sizes and more sensitive powers,and even for“large p small n”.Then this paper proposes a new test for the equality of the two-sample covariance matrices,which refers to the form of corrected Rao's score test statistic,and by the central limit theorem for lin-ear spectral statistic of high-dimensional random F-matrices proposed by Zheng(2012),this paper proves asymptotic normality property as the dimension p and sample size(n1,n2)tend to infinity.Because the theorem proposed by Zheng(2012)includes the asymptotic normality property under the non-Gaussian assumption and establishes the 4th moment of variable which is limited,so the new test has a wide range for non-Gaussian and without the 4th moment restriction.Finally simulation study shows that the new test statistic is made suitable for high-dimensional Gaussian variables and non-Gaussian variables.