Simultaneous Testing Of Mean Vector And Covariance Matrix For High-dimensional Population

With the rapid development of science and technology,it brought about a large number of high-dimensional datas,e.g.,gene expression spectrum datas,single nucleotide polymorphisms datas,online consumer price datas,stock datas,etc(see Lam and Yao(2012),Chang et al.(2015)and others).The emergence of high-dimensional datas brought about challenge for many classic statistical methods and theories.An important feature of high-dimensional datas is the dimension of datas p much larger than sample size n.The classical multivariate statistical analysis is invalid under high-dimensional case.Because the classical multivariate statistical analysis is based on the assumption that p is less than n.So the emergence of high-dimensional datas brings an urgent need for us to update and rewrite some traditional multivariate analysis methods.Mean vector and covariance matrix are two basic parameters in multivariate statistical analysis,which are used to characterize the population.Checking the structure of the population mean vector and the population covariance matrix has been one of the focuses of statistical analysis.In this paper,we will focus on three hypothesis tests for the high-dimensional population mean vector and covariance matrix.First,for high-dimensional data,a new test procedure is proposed to simultaneously test mean vector and covariance matrix being equal to a given vector and a given positive definite matrix respectively.It allows the case of large dimension p and small sample size n,and it is also robust against non-normal data.Besides,the asymptotic null distribution is derived and the asymptotic theoretical power function is explicitly achieved.The local power of the new method is studied and the proposed test is proved to be asymptotically unbiased.The efficiency of the new method is assessed by numerical simulations.Second,a new test procedure is proposed to simultaneously test mean vector and covariance matrix of one high-dimensional data being equal to mean vector and covariance matrix of another high-dimensional data respectively.It allows the case of large dimension p and small sample size n,and it is valid for both normal data and non-normal data.Besides,the asymptotic null distribution is derived and the asymptotic theoretical power function is explicitly achieved.The proposed test is proved to be asymptotically unbiased.The efficiency of the new method is assessed by numerical simulations.Finally,we propose a new test procedure for checking the population covariance matrix to have an intraclass correlation pattern under a high-dimensional setting.This new test procedure is valid for both normal data and non-normal data.Based on the martingale difference theory,the asymptotic normality of the test statistic is studied when the sample size and the dimension increase proportionally.And we proof that the new test statistics is consistent.The efficiency of the new method is assessed by numerical simulations.