Empirical Likelihood Method For Testing Mean And Covariance Matrix Of High-dimensional Data

With the arrival of big data era,both the size and the dimension of the data become larger and larger,it is the phenomenon of "large p,large n" discussed in the statistical literature.However,the classical statistical analysis method is no longer applicable on studying the "large p,large n"type data.Therefore,it is a meaningful work to seek some appropriate and effective statistical analysis methods to deal with high-dimensional data.In the paper,we mainly adopt nonparametric empirical likelihood method and statistical asymptotic theory to study the hypothesis test for the covariance matrix in high-dimensional transposable matrix-valued data with kronecker product dependence structure.At the same time,we will also consider the test on the linear combination of the mean value vectors of two population with fixed dimension.For the identity test on the covariance matrix of the matrix-valued data,under the column independence assumption,Touloumis et al.[4]adopt the U statistics method proposed by Chen et al.[6]to consider the sphericity and identity test on the assumed nonparametric model,they proved the test statistics obey normal distribution asymptotically.In the paper,we will construct the estimation equation by straightening the random matrix,and use the empirical likelihood method to test the covariance matrix of this matrix-valued data,it is proved that likelihood ratio statistic asymptotical Chi-square distributed.Some numerical simulation show the efficiency of the proposed method.For the test problem on the linear combination of two sample mean value vectors,a re-markable work due to Li et al.[31],who study the linear combination of mean vector of high-dimensional data by constructing U statistics.In the paper,we will adopt four types of slightly different empirical likelihood methods to revisit the test problem,it is concluded that all the likelihood ratio statistics asymptotically obey Chi-square distribution.At the same time,some simulations are conducted by the bootstrap trick,the advantages and disadvantages of each empirical likelihood method are analysed.